Hardy inequalities with double singular weights
aa r X i v : . [ m a t h . A P ] S e p Hardy inequalities with double singular weights
Nikolai Kutev ∗ Tsviatko Rangelov ∗ Contents γ p . . . . . . . . . . . . . . . . . . . . . . 433.2 Examples and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 λ p,n using Hardy inequalities . . . . . . . . . . . . 716.3 Comparison between different analytical estimates of λ p,n ( B R ) . . . . . . . 83 ∗ Institute of Mathematics and Informatics, Bulgarian Academe of Sciences, 1113, Sofia, BulgariaCorresponding author: T. Rangelov, [email protected] bstract The aim of this paper is to obtain new Hardy inequalities with double singularweights – at an interior point and on the boundary of the domain. These inequalitiesgive us the possibility to derive estimates from below of the first eigenvalue of thep-Laplacian with Dirichlet boundary conditions.
Keywords:
Hardy inequality; Double singular weights; Sharp estimates; p-Laplacian;First eigenvalue.
The paper is devoted to the classical Hardy inequality, its generalizations and applicationsfor estimates from below of the first eigenvalue λ p,n (Ω) of the p-Laplacian, p >
1, ina bounded domain Ω ⊂ R n , n ≥
2. Only the multidimensional case n ≥ n = 1 there exist detailed literature and satisfactory results, see for instanceHardy [55, 56, 57], Neˆcas [83], Maz’ja [80], Opic and Kufner [84], Hoffmann-Ostenhof et al.[59]. Unlike the one-dimensional case, the theory for n ≥ ∂ Ω, we always prove the optimality of Hardy constant. As for the sharpness of the Hardyinequality, we show that an equality is achieved only for Hardy inequality with additional‘nonlinear’ term. It is well-known that for inequalities with optimal constant and addi-tional ‘linear’ term, an equality is not achieved. Only in the case when Hardy constantis greater than the optimal one, the sharpness of the inequality is proved by variationaltechnique, see for example Pinchover and Tintarev [87] and the references therein. In fact,in this way the optimality of Hardy constant is shown.In the literature mainly inequalities with singular weights at a point, or on the bound-ary, ∂ Ω, or on some k -dimensional manifold, 1 ≤ k ≤ n −
1, have been studied. Thesubject of the investigations in the paper is Hardy type inequalities in bounded domainsΩ ∈ R n , n ≥ ∂ Ω.Our aim is to derive new Hardy inequalities, which are with an optimal constant andwith suitable additional terms that become sharp. For example, Hardy constant is optimalfor convex and star-shaped domains and the inequality is sharp as a result of a ‘nonlinear’additional term.The background of the theory of Hardy inequalities is mathematical and functionalanalysis and differential equations. Among many different applications we choose one –the estimate of the first eigenvalue of the p-Laplacian from below, which motivates thestudy of Hardy inequalities with double singular weights.2here are estimates for λ p,n (Ω) by means of the Cheeger’s constant, Cheeger [27],Lefton and Wei [71], Kawohl and Fridman [63], by the Picone’s identity Benedikt andDr´abek [18, 19], with the Sobolev inequality Maz’ja [80], Ludwig et al. [79], with estimatesin parallelepiped Lindqvist [78] and others. However, Hardy inequality with double sin-gular weights allows one to get better analytical estimates for λ p,n (Ω). For completeness,we prove estimates from below for λ p,n (Ω) as well as by the well-known Hardy inequalitieswith singular weights only at a point or on the boundary ∂ Ω. The comparison of theresults definitely shows that the inequalities with double singular weights produce bet-ter analytical estimates for λ p,n (Ω) in comparison with those obtained from other Hardyinequalities or other methods.In the rest of this section, without claims of completeness, we present results of Hardyinequalities which from our point of view are decisive for the development of the subjectin the recent years, as the books of Maz’ja [80], Opic and Kufner [84], Ghoussoub andMoradifam [51], Balinsky et al. [12], Kufner et al. [67], and the papers of Brezis and Marcus[25], Brezis and Vazquez [26], Davies [28], Barbatis et al. [13, 14], Hoffmann-Ostenhof et al.[59], Tidblom [94], Edmunds and Hurri-Syrj¨anen [31], Kinnunen and Korte [64], Lehrb¨ack[73] and others. In section 2 we derive a new Hardy inequality with weights in abstractform. Particular cases are presented to demonstrate the applicability of the method andto show some generalizations of the existing results. In section 3 we prove a general Hardyinequality with singular weights at zero and on the boundary ∂ Ω of star-shaped domainsand an optimal Hardy constant. In section 4 we propose Hardy inequality with weightsingular at 0 ∈ Ω in the class of functions which are not zero on the boundary ∂ Ω. TheHardy constant is optimal and the inequality is sharp due to the additional boundary term.In section 5 we derive improved Hardy inequality with double singular weights in bounded,star-shaped domains Ω ⊂ R n , n ≥ In this section classical Hardy inequalities are shown together with some definitions used inthe paper. The analysis of the existing in the literature results is far from its completeness.The aim of this section is to recall the well known results in the frame of the study in thepaper.The classical Hardy inequality in R = (0 , ∞ ) states Z ∞ | u ′ ( x ) | p x α dx ≥ (cid:18) p − − αp (cid:19) p Z ∞ x − p + α | u ( x ) | p dx, (1.1)where 1 < p < ∞ , α < p − u ( x ) is an absolutely continuous function on [0 , ∞ ) with u (0) = 0, see Hardy [56, 57] for α = 0 and Hardy et al. [58], Sect. 9.8 for α < p − (cid:18) p − − αp (cid:19) p is the best possible one, i.e., it can not be replaced witha greater one, but the equality in (1.1) is not achieved.In the last 20 years the generalizations of (1.1) for the multidimensional case are mainlyoriented in two directions with respect to the structure of the singular weight:3 singularities on the boundary: when the prototype inequality is Z Ω |∇ u ( x ) | p d α ( x ) dx ≥ C Ω Z Ω d − p + α ( x ) | u ( x ) | p dx, (1.2)with d ( x ) = dist( x, ∂ Ω) and α < p − p > n ≥ p = n , u ∈ W ,p (Ω); • singularity at a point: when the inequality Z Ω |∇ u ( x ) | p dx ≥ C p,n Z Ω | u ( x ) | p | x | p dx, (1.3)holds, Ω ⊂ R n , 0 ∈ Ω, n ≥ p > p = n , u ∈ W ,p (Ω) for p < n and u ∈ W ,p (Ω \{ } ) for p > n . The constant C p,n = (cid:12)(cid:12)(cid:12)(cid:12) n − pp (cid:12)(cid:12)(cid:12)(cid:12) p is optimal one.We will present some of the results on Hardy inequalities underlining their optimality andsharpness, see Definition 1.1Next we recall several useful definitions and notions. Definition 1.1.
The constants C Ω in Hardy inequality Z Ω V ( x ) |∇ u | p dx ≥ C Ω Z Ω W ( x ) | u | p dx + A ( u ) (1.4) with positive weights V ( x ) , W ( x ) and additional nonnegative term A ( u ) is optimal if forevery ε > there exists u ε from the admissible class of functions for which the inverseHardy inequality holds if we replace C Ω with C Ω + ε . The Hardy inequality (1.4) is sharp ifthere exists a function from the admissible class of functions for which (1.2) is an equalityand both sides of (1.2) are finite. Let us recall that sharp Hardy inequalities are proved by means of variational tech-nique, see Pinchover and Tintarev [87] and the references therein.
Inequalities with general weights
There are generalizations of (1.1) for the n -dimensional case, n ≥
2, for bounded domainsand for different weights (integral kernels), see for more details Davies [28], Opic andKufner [84], Ghoussoub and Moradifam [51], Balinsky et al. [12].For example, in Opic and Kufner [84], Theorems 14.1, 14.2, the following Hardy in-equality with general weights is proposed n X i =1 Z Ω v i ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂x i (cid:12)(cid:12)(cid:12)(cid:12) p dx ≥ Z Ω w ( x ) | u ( x ) | p dx. (1.5)Here Ω is a bounded domain in R n , n ≥ u ∈ C ∞ (Ω), 1 < p < ∞ , functions v i , i = 1 , . . . , n and w are measurable, positive and finite for a. e. x ∈ Ω. It is proved thatunder the existence of a solution y ( x ) of the equation n X i =1 ∂∂x i " v i (cid:12)(cid:12)(cid:12)(cid:12) ∂y∂x i (cid:12)(cid:12)(cid:12)(cid:12) p − ∂y∂x i + w ( x ) | y | p − y = 0 in Ω , (1.6)4here y ( x ) = 0 and ∂y∂x i = 0 for a. e. x ∈ Ω, and some regularity conditions on y , v i and w inequality (1.5) hold.A necessary and sufficient condition on v i and w for the validity of (1.5) is proved inMaz’ja [80] in terms of capacities, see also Opic and Kufner [84], Theorem 16.3 and thediscussion therein.In Ghoussoub and Moradifam [51], Theorem 4.1.1, Hardy inequality with general pos-itive weights V ( | x | ) and W ( | x | ) is proposed in the ball B R = { x ∈ R n , | x | < R } , n ≥ Z B R V ( | x | ) |∇ u | dx ≥ c Z B R W ( | x | ) u dx, u ∈ C ∞ ( B R ) . (1.7)The necessary and sufficient condition for the validity of (1.7) given in the same book isthat the couple ( V, W ) forms n-dimensional Bessel pair on the interval (0 , R ), i.e., theequation y ′′ ( r ) + (cid:18) n − r + V ′ ( r ) V ( r ) (cid:19) y ′ ( r ) + W ( r ) V ( r ) y ( r ) = 0 , has a positive solution in (0 , R ).Another Hardy inequality is proved in Shen and Chen [92], Lemma 1.1, Z Ω φ ( r ) |∇ u | dx ≥ Z Ω φ ( r ) (cid:12)(cid:12)(cid:12)(cid:12) h ′ ( r ) h ( r ) (cid:12)(cid:12)(cid:12)(cid:12) u dx, r = | x | , (1.8)in a bounded domain Ω ⊂ R n , 0 ∈ Ω ⊂ B R , R > sup x ∈ Ω | x | . Here φ ( r ) ∈ C (0 , R ), φ ( r ) > h ∈ C (0 , R ) is a positive solution of the equation r n − φ ( r )( h ) ′ ( r ) = c = const , (1.9)and u ∈ C ∞ (Ω) if h − (0) = 0 while u ∈ C ∞ (Ω \{ } ) if h − (0) = 0.In Davies and Hinz [29], see also Balinsky et al. [12], Theorem 1.2.8, the followingHardy inequality is shown Z Ω |∇ V | p | ∆ V | p − |∇ u | p dx ≥ (cid:18) p (cid:19) p Z Ω | ∆ V || u | p dx, for all u ∈ C ∞ (Ω) and any domain Ω ⊂ R n , n ≥
2. The real valued weight V ( x ) hasderivatives up to order 2 in L loc (Ω) and ∆ V is of one sign for a.e. x ∈ Ω. Inequalities with weights singular on the boundary
Another research direction on Hardy inequalities concerns the study of the geometricproperties of the domain Ω ⊂ R n , on which the Hardy’s inequality (1.2) holds.The inequality (1.2) was proved by Neˆcas [83] for bounded domains Ω with Lipschitzboundary ∂ Ω and u ∈ C ∞ (Ω). Next generalizations of (1.2) were made by Kufner [66] forH¨older ∂ Ω and by Wannebo [100] for ∂ Ω under generalized H¨older conditions. Detaileddescription of these results can be found in Opic and Kufner [84], Maz’ja [80], Hajlasz [54].Another geometric condition on ∂ Ω for the validity of (1.2) is suggested in Edmundsand Hurri-Syrj¨anen [31]. More precisely, if R n \ Ω is b -plump for some b , then (1.2) holdsfor α ≤
0. If additionally Ω also satisfies the Witney cube counting condition then (1.2)5s valid for α ∈ (0 , α ), where α > t = { x ∈ Ω , d ( x ) < t } for u ∈ C ∞ (Ω) and without zero conditions for u on the set { x ∈ Ω , d ( x ) = t } .A more generally sufficient condition result is proved in Koskela and Lehrb¨ack [65].Another way to describe the properties of Ω is to connect the validity of inequality (1.2)with the existence of solutions of certain boundary value problems for second order ellipticequation with a singular weight. In Ancona [7] it is proved that a necessary and sufficientcondition for the validity of (1.2) when p = 2 and α = 0 is the existence of a positivesuper–harmonic function v in Ω and a positive number δ such that ∆ v + δd ( x ) v ≤ δ = C Ω . For more general results for p-Laplaceoperator see Allegretto and Huang [5].For arbitrary p > α = 0 in Kinnunen and Korte [64], Theorem 5.2, it is provedthat (1.2) holds if and only if there exists a positive constant λ p (Ω) and a positive super-solution ϕ ∈ W ,p (Ω) of the problemdiv (cid:0) |∇ ϕ | p − ∇ ϕ (cid:1) + λ p | ϕ | p − ϕd p ( x ) ≤ , ϕ = 0 on ∂ Ω . The constant C Ω = (cid:18) p − − αp (cid:19) p is optimal for (1.2) in the case n = 1 and for convexdomains for n ≥
2. For non-convex domains when n > C Ω in (1.2)is unknown. There are only partial results, for example, when Ω is non-convex and p = 2and α = 0, then Ancona [7] proved that C Ω ≥
116 by means of the Koebe quarter theorem.Later on in Laptev and Sobolev [70] better estimate for C Ω using a stronger version ofKoebe quarter theorem was obtained.Moreover, in Avkhadiev [10] a sufficient condition on non-convex domain Ω is given sothat (1.2) with p = 2 , α = 0 holds with optimal constant C Ω = 14 for n ≥ C smooth boundary ∂ Ω with non-negative meancurvature H ( x ) inequality (1.2) is proved in Lewis et al. [76], see Theorem 1.2, for α = 0with optimal constant C Ω = (cid:18) p − p (cid:19) p . The curvature condition H ( x ) ≥ ε > H ( x ) ≥ − ε on ∂ Ω, the Hardy’s inequality (1.2) with C Ω = (cid:18) p − p (cid:19) p fails for some u ∈ W ,p (Ω).Recently, in Zsupp´an [101], Theorem 2.1, an explicit estimate from below for C Ω in(1.2) for star-shaped domains Ω was shown. This estimate coincides with (cid:18) p − − αp (cid:19) p for convex domains Ω.For the annular domain B R \ B r = { < r < | x | < R < ∞} ⊂ R n the following Hardy6nequality is proved in Avkhadiev and Laptev [9], see Theorem 1 and Corollary 1, Z B R \ B r |∇ u | dx ≥ Z B R \ B r (cid:18) ( n − n − | x | + 1 | x − r | + 1 | x − R | + 2 | x − r || x − R | (cid:19) u dx, for every u ∈ H ( B R \ ¯ B r ). The weights on the right-hand side are singular on the boundaryof B R \ ¯ B r and at the origin, which however does not belong to the domain B R \ ¯ B r . Inequalities with weights singular at a point
Another generalization of (1.1) is an inequality with a weight, singular at an interior pointof Ω ⊂ R n , i.e., namely of type (1.3). The optimal constant C ,n = (cid:18) n − (cid:19) is obtainedin Leray [74] for Ω = R , p = 2 and in Hardy et al. [58] for Ω = R n , n > p = 2, see alsoPeral and Vazquez [86] and Opic and Kufner [84].The case p = n in (1.3) is considered in Ioku and Ishiwata [61], see Theorem 1.1, whereHardy’s inequality Z B (cid:12)(cid:12)(cid:12)(cid:12) h x | x | , ∇ u i (cid:12)(cid:12)(cid:12)(cid:12) n dx ≥ (cid:18) n − n (cid:19) n Z B | u | n | x | n (cid:16) log | x | (cid:17) n dx, is proved with optimal constant (cid:18) n − n (cid:19) n for every u ∈ W ,n ( B ).For function u ∈ C ∞ (Ω), the constant C p,n in (1.3) is independent on Ω. However,when u ∈ C ∞ (Ω) the boundary term on ∂ Ω is taken into account because u does notnecessary vanish on the boundary and the geometry of Ω is important. In Wang and Zhu[98], for p = 2, the following Hardy inequality with weights and an additional boundaryterm was proposed. Z B | x | − α |∇ u | dx ≥ (cid:18) n − − α (cid:19) Z B | x | − α +1) u dx − n − α − Z ∂B u dx, for B ⊂ R n , n ≥ α < n −
22 .In Kufner [66] Hardy inequality for functions which are not zero on the boundary (orpart on the boundary) is considered. In this case a natural boundary term is added to theright-hand side.
When Hardy’s constant in (1.2) is optimal there is no non-trivial function of the admissibleclass of functions for which the Hardy inequality becomes an equality.That is why in Brezis and Marcus [25] the question on the existence of an additionalpositive term A ( u ) to the right-hand side of (1.2) is stated such that the improved in-equality Z Ω |∇ u ( x ) | p d α ( x ) dx ≥ C Ω Z Ω | u ( x ) | p d α − p ( x ) dx + A ( u ) , (1.10)7till holds in bounded domains for the optimal constant C Ω = 14 when p = 2, α = 0, n ≥ u ∈ C ∞ (Ω) Z Ω |∇ u | dx − Z Ω u d ( x ) dx ≥ b (Ω) Z Ω u dx, (1.11)where b (Ω) ≥ (Ω) and diam(Ω) = max x,y ∈ Ω | x − y | . In Hoffmann-Ostenhof et al. [59], Theorem 3.2, for p = 2, n ≥ C Ω = 14 and a convexdomain Ω, inequality (1.11) is improved by showing that b (Ω) ≥ b | Ω | − n , b = n " π n n Γ( n/ n , (1.12)while in Evans and Lewis [33], Theorem 3.2, the estimate (1.12) is improved for boundedconvex domain Ω ⊂ R n to b (Ω) ≥ b | Ω | − n .Later on in Tidblom [94], Theorem 2.2, for p > n ≥ α = 0 an optimal constant C Ω = (cid:18) p − p (cid:19) p is shown for inequality (1.10), which holds with A ( u ) = b (Ω) Z Ω | u | p dx where b (Ω) = ( p − p +1 p p (cid:18) ω n n | Ω | (cid:19) p/n √ π Γ (cid:0) n + p (cid:1) Γ (cid:16) p +12 (cid:17) Γ (cid:0) n (cid:1) . In Filippas et al. [44], Theorem 1.1, the authors proved that in a convex, boundeddomain Ω ⊂ R n inequality (1.11) holds with b (Ω) ≥ D int (Ω)) − , D int (Ω) = 2 sup x ∈ Ω d ( x ) . Moreover, for 1 < p < n , α = 0, and C Ω = (cid:18) p − p (cid:19) p , the estimate (1.10) holds with A ( u ) = b (Ω) Z Ω | u | p dx and C ( p, n ) D − pint ≥ b (Ω) ≥ C ( p, n ) D − pint , for some positive constants C ( p, n ) , C ( p, n ).A different estimate for b (Ω) in (1.11) is given in Avkhadiev and Wirths [11], Theorem1, where b (Ω) = b D int , and b ≈ . . . . is the Lamb constant defined as the first positive zero of the function J ( x ) + 2 xJ ′ ( x ), where J is the Bessel function of order 0.Another additional term for (1.10) is proposed in Brezis and Marcus [25], Theorem5.1, for convex domains A ( u ) = 14 Z Ω u d ( x ) (1 − log( d ( x ) /L )) dx, L = diam Ω . p > p = n and in domain Ω satisfying some geometric condition, like the meanconvexity of Ω, the result of Brezis and Marcus [25] is generalized in Barbatis et al. [14],Theorem A, to Hardy inequality Z Ω |∇ u | p dx ≥ (cid:18) p − p (cid:19) p Z Ω | u | p d p ( x ) dx + 12 (cid:18) p − p (cid:19) p − Z Ω | u | p d p ( x ) (log( d ( x ) /D )) − dx, (1.13)for u ∈ W ,p (Ω) and D ≥ sup x ∈ Ω d ( x ).In Filippas et al. [44], Theorem 3.1, for a convex domain Ω ⊂ R n , 1 < p < n , α > − p and C Ω = (cid:18) p − p (cid:19) p the following Hardy inequality with additional term is proved Z Ω |∇ u | p dx ≥ (cid:18) p − p (cid:19) p Z Ω | u | p d p ( x ) dx + C ( p, n, α ) D − α − pint Z Ω d α ( x ) | u | p dx. The constant C ( p, n, α ) is independent of Ω and for p = 2 , α > − C (2 , n, α ) = C α is given explicitly: C α = 2 α ( α + 2) when − < α < −
1, while C α = 2 α (2 α + 3) when α ≥ − ⊂ R n +1 with C smooth boundary ∂ Ω with nonnegative mean curvature H ( x ) ≥ Z Ω |∇ u | dx ≥ Z Ω u d ( x ) dx + b ( n, Ω) Z Ω u dx, is proved where b ( n, Ω) ≥ n (cid:18) inf ∂ Ω H ( x ) (cid:19) .In Edmunds and Hurri-Syrj¨anen [31], Theorem 5.1, for the domain Ω ⊂ R n which is b -plump for some b ∈ (0 , Z Ω |∇ u | p dx ≥ c ( p, n ) b n (cid:18)Z Ω | u | p d p ( x ) dx + | Ω | − p/n Z Ω | u | p dx (cid:19) , is satisfied for every u ∈ W ,p (Ω), 1 < p < ∞ . The constant c ( p, n ) is given explicitly.For arbitrary C smooth domains Ω ⊂ R n , it has been established in Psaradakis [90],Theorem A, for the limiting case p = 1 the inequality Z Ω |∇ u | d s − ( x ) dx ≥ ( s − Z Ω | u | d ( x ) dx + B Z Ω | u | d ( x ) dx, (1.14)for s ≥ B ≥ ( n − H ( x ) and every u ∈ C ∞ (Ω), where H ( x ) ≥ ∂ Ω. If s ≥ L Hardy inequalities with weights, i.e., p = 1 in (1.10), see Psaradakis[88]. 9et us note the paper of Frank and Loss [48] where for inequality in a domain withgeneralized distance to the boundary a reminder term with the Sobolev-critical exponentis derived . In the particular case of a convex domain this settles a conjecture by Filippaset al. [45].Analogously to the paper of Brezis and Marcus [25] in Brezis and Vazquez [26] thequestion about the existence of an additional positive term A ( u ) such that the improvedinequality Z Ω |∇ u ( x ) | dx ≥ (cid:18) n − (cid:19) Z Ω u ( x ) | x | dx + A ( u ) , (1.15)with optimal constant still holds for every u ∈ H (Ω) is addressed. In the same paper,Theorem 4.1, the authors find A ( u ) = λ (Ω) Z Ω u ( x ) dx, λ (Ω) = z (cid:18) ω n | Ω | (cid:19) /n , where z ≈ . J ( z ) whereas ω n and | Ω | arethe volume of the unit ball and resp. Ω.The following generalization of (1.15) for 1 < p < n is proved in Gazzola et al. [49],Theorem 1, Z Ω |∇ u | p dx ≥ (cid:18) n − pp (cid:19) p Z Ω | u | p | x | p dx + C ( p, n ) (cid:18) ω n | Ω | (cid:19) pn Z Ω | u | p dx, (1.16)where Ω is a bounded domain, 0 ∈ Ω, the constant (cid:18) n − pp (cid:19) p is optimal and C ( p, n ) isgiven explicitly for p ≥ Z Ω |∇ u | dx ≥ (cid:18) n − (cid:19) Z Ω u | x | dx + 14 ∞ X i =1 Z Ω u | x | X (cid:18) | x | R (cid:19) . . . X i (cid:18) | x | R (cid:19) dx, for every u ∈ H (Ω). Here R ≥ sup x ∈ Ω | x | , X ( t ) = (1 − ln t ) − , X k ( t ) = X ( X k − ( t )) for k = 2 , . . . , p = 2, n ≥ ⊂ R n is a bounded domain, 0 ∈ Ω.When u ∈ W ,p (Ω) is not zero on ∂ Ω, the result in Filippas and Tertikas [42] isextended in Adimurthi and Esteban [2], Theorem 1.1, to the inequalities Z Ω |∇ u | p dx ≥ (cid:18) n − pp (cid:19) p Z Ω | u | p | x | p dx + C ( p, n ) Z Ω k X j =1 (cid:16) log ( j ) ( R/ | x | ) (cid:17) − p | u | p | x | p dx + b (Ω , p, R ) Z ∂ Ω | u | p ds, for 1 < p < n, Z Ω |∇ u | n dx ≥ (cid:18) n − n (cid:19) n Z Ω | u | n ( | x | log R/ | x | ) n dx + C ( n ) Z Ω k X j =2 (cid:16) log ( j ) ( R/ | x | ) (cid:17) − n | u | n | x | n dx + b (Ω , n, R ) Z ∂ Ω | u | n ds, for p = n. Here Ω ⊂ R n is a bounded domain, 0 ∈ Ω, log (1) a = log a , log ( k ) a = log(log ( k − a )with a > e ( k − , k ≥
2, log ( k ) a = k Y j =1 log ( j ) a for a > e ( k − , e (1) = e , e ( k +1) = e e k and R > e ( k − sup x ∈ Ω | x | .Let us mention the result of Wang and Willem [97], Theorem 2, where the weights inboth sides of Hardy inequality (1.15) are singular Z Ω | x | α |∇ u | dx ≥ (cid:18) n − α (cid:19) Z Ω u | x | − α dx + 14 Z Ω u | x | − α ln − R | x | dx, and R > sup x ∈ Ω | x | , α > − n . The constants (cid:18) n − α (cid:19) and 14 are optimal.Another direction proposed in V´azquez and Zuazua [95], Theorem 2.2, is the improvedHardy–Poincare inequality Z Ω |∇ u | dx ≥ (cid:18) n − (cid:19) Z Ω u | x | dx + C ( q, Ω) (cid:18)Z Ω | u | q dx (cid:19) /q , for 1 ≤ q <
2, Ω ⊂ R n is a bounded domain, 0 ∈ Ω and u ∈ H (Ω).In Filippas and Tertikas [42], Theorem A, and in Adimurthi et al. [4], Theorem A,the following result for the improved Hardy–Sobolev inequality in the unit ball B ⊂ R n , n ≥ Z B |∇ u | dx ≥ (cid:18) n − (cid:19) Z B | u | | x | dx + C n ( a ) (cid:18)Z B X n − n − ( a, | x | ) | u | nn − dx (cid:19) n − n , (1.17)for every u ∈ C ∞ ( B ), where X ( a, s ) = ( a − log s ) − , a > , < s <
1. Here (cid:18) n − (cid:19) and C n ( a ) are optimal constants, where C n ( a ) is given by C n ( a ) = ( n − n − n S n , a ≥ n − ,a n − n S n , < a < n − , (1.18)and S n = πn ( n − (cid:16) Γ (cid:16) n (cid:17) / Γ( n ) (cid:17) n is the best constant in the classical Sobolev inequality Z R n |∇ u | dx ≥ S n (cid:18)Z R n | u | nn − dx (cid:19) n − n .
11n improvement of Hardy inequality (1.15) and (1.17) for p > n , which is of independentinterest, is proved in Psaradakis [89], Theorem C.In Barbatis et al. [14], Theorem A, the improved Hardy inequality Z Ω |∇ u | p dx ≥ (cid:12)(cid:12)(cid:12)(cid:12) n − pp (cid:12)(cid:12)(cid:12)(cid:12) p Z Ω | u | p | x | p dx + p − p (cid:12)(cid:12)(cid:12)(cid:12) n − pp (cid:12)(cid:12)(cid:12)(cid:12) p − Z Ω | u | p | x | p X (cid:18) | x | D (cid:19) dx, (1.19)is proved in a bounded domain Ω ⊂ R n with 0 ∈ Ω and u ∈ W ,p (Ω \{ } ).Here X ( t ) = − / log t , t ∈ (0 ,
1) and D ≥ D where D ( n, p ) ≥ sup x ∈ Ω | x | is some positiveconstant. The constants in (1.19) are optimal. In fact, (1.13) and (1.19) are a consequenceof a more general result in Barbatis et al. [14], Theorem A, when the distance function d ( x ) is the distance of x ∈ Ω to a piecewise smooth surface K of the co-dimension k ,1 ≤ k ≤ n .For p > n improvements of Hardy inequality (1.15) are given in Psaradakis [89]: The-orem A for Sobolev-type inequality and Theorem B for Hardy-Morrey inequality.Let us mention the paper Frank [47] where for the Hardy inequality with a pointsingularity a reminder term involving fractional power of the Laplacian is derived. By theembedding theorems, this implies inequalities in the spirit of Brezis and Vazquez [26] aswell remainder terms with L q norms.Unfortunately, in all papers mentioned above, inequality (1.15) with optimal constant (cid:18) n − (cid:19) or (1.16) with optimal constant (cid:18) n − pp (cid:19) p are not sharp, see Definition 1.1.In fact, in Pinchover and Tintarev [87] (see also the references therein) it was shown byvariational technique that Hardy inequality (1.3) with p = 2 is not sharp for the optimalconstant (cid:18) n − (cid:19) .Finally, we will briefly refer to the case of a point singularity of the weights on theboundary ∂ Ω, i.e., 0 ∈ ∂ Ω, see Filippas and Tertikas [43], Fall and Musina [41], Barbatiset al. [16], Devyver et al. [30] and the references therein. In this case the optimal constant (cid:18) n − (cid:19) in (1.15) for the weight singular at an interior point is replaced with greaterconstant n In this section we derive a new Hardy inequalities with weights in abstract form. Examplesare presented to demonstrate the applicability of the method and to show generalizations ofexisting results. The sharpness of the inequalities is proved and the results are illustratedby several examples. The section is based on Fabricant et al. [36, 37].
Let Ω be a bounded domain, Ω ⊂ R n , n ≥ ∂ Ω ∈ C . Suppose that f isa vector function defined in Ω, | f | 6 = 0 with components f i ∈ C (Ω) ∩ C ( ¯Ω), i = 1 , · · · , n .Let p > v , w , v − p ∈ L (Ω) suchthat 12 div f − ( p − v | f | p ′ ≥ w, for a.e. x ∈ Ω , (2.1)where 1 p + 1 p ′ = 1. Let ∂ Ω be divided into two parts ∂ Ω = Γ − ∪ Γ + , whereΓ − = { x ∈ ∂ Ω : h f, η i < } , Γ + = { x ∈ ∂ Ω : h f, η i ≥ } . (2.2)Here η is the unit outward to Ω normal vector on ∂ Ω and h ., . i is the scalar product in R n .We consider the functions u ∈ C ∞ Γ − (Ω), where C ∞ Γ − = { u ∈ C ∞ , u = 0 in a neighbourhood of Γ − } . Let us introduce the notations L ( u ) = Z Ω v − p (cid:12)(cid:12)(cid:12)(cid:12) h f, ∇ u i| f | (cid:12)(cid:12)(cid:12)(cid:12) p dx, K ( u ) = Z Γ + h f, η i| u | p dS,K ( u ) = Z Ω v | f | p ′ | u | p dx, N ( u ) = Z Ω w | u | p dx, (2.3)where dS is the (n-1)-dimensional surface measure and u ∈ C ∞ Γ − (Ω).In this section our main result is the following theorem. Theorem 2.1.
Under condition (2.1) for every u ∈ C ∞ Γ − (Ω) , u , and v > , w ≥ ,the following inequality holds L ( u ) ≥ (cid:18) p (cid:19) p ( K ( u ) + ( p − K ( u ) + N ( u )) p K p − ( u ) . (2.4) Proof.
Since Z Ω h f, ∇| u | p i dx = p Z Ω | u | p − u h f, ∇ u i dx, applying the H¨older inequality on the right-hand side with v − /p ′ h f, ∇ u i| f | and v /p ′ | f || u | p − u, as factor of the integrand we get (cid:12)(cid:12)(cid:12)(cid:12)Z Ω h f, ∇| u | p i dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ p (cid:18)Z Ω v − p (cid:12)(cid:12)(cid:12)(cid:12) h f, ∇ u i| f | (cid:12)(cid:12)(cid:12)(cid:12) p dx (cid:19) /p (cid:18)Z Ω v | f | p ′ | u | p dx (cid:19) /p ′ . (2.5)Rising both sides of (2.5) to power p it follows that Z Ω v − p (cid:12)(cid:12)(cid:12)(cid:12) h f, ∇ u i| f | (cid:12)(cid:12)(cid:12)(cid:12) p dx ≥ (cid:12)(cid:12)(cid:12)(cid:12) p Z Ω h f, ∇| u | p i dx (cid:12)(cid:12)(cid:12)(cid:12) p (cid:18)Z Ω v | f | p ′ | u | p dx (cid:19) p − . (2.6)13ntegrating by parts the numerator of the right-hand side of (2.6), from (2.1), we get (cid:12)(cid:12)(cid:12)(cid:12) p Z Ω h f, ∇| u | p i dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) p Z ∂ Ω h f, η i| u | p dS − p Z Ω div f | u | p dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) p Z ∂ Ω h f, η i| u | p dS − p Z Ω (div f + ( p − v | f | p ′ ) | u | p dx + (cid:18) p − p (cid:19) Z Ω v | f | p ′ | u | p dx (cid:12)(cid:12)(cid:12)(cid:12) ≥ p (cid:12)(cid:12)(cid:12)(cid:12)Z ∂ Ω h f, η i| u | p dS + Z Ω w | u | p dx + ( p − Z Ω v | f | p ′ | u | p dx (cid:12)(cid:12)(cid:12)(cid:12) ≥ p (cid:18)Z Γ + h f, η i| u | p dS + Z Ω w | u | p dx + ( p − Z Ω v | f | p ′ | u | p dx (cid:19) . The last equality follows from u (cid:12)(cid:12) Γ − = 0.From (2.6) we obtain (2.4). Remark 2.1.
The idea of the proof of Theorem 2.1 comes from Boggio [24], for p = 2;Flekinger et al. [46], Theorem II.1 and Barbatis et al. [14], Theorem 4.1. In our case, incontrast to these works, we consider functions not necessarily zero on the whole boundary ∂ Ω and due to this there is an additional boundary term K in (2.4). In L and K thereis also a weight v , which is 1 in the above mentioned papers.The careful analysis of the proof of Theorem 2.1 shows that we have a more generalresult than (2.4) without any sign conditions of the boundary term. Corollary 2.1.
Suppose that p > and there exist in Ω measurable functions v > , w, v − p ∈ L (Ω) such that condition (2.1) holds. Then for every u ∈ C ∞ ( ¯Ω) the followinginequality holds L ( u ) ≥ (cid:18) p (cid:19) p | K ( u ) + ( p − K ( u ) + N ( u ) | p K p − ( u ) , (2.7) where K ( u ) = Z ∂ Ω h f, η i| u | p dS . Corollary 2.2.
As a consequence of Theorem 2.1 we get under condition (2.1) for v > , w ≥ , v − p ∈ L (Ω) and u ∈ C ∞ Γ − (Ω) the following Hardy inequalities:i) L p ( u ) ≥ (cid:18) p ′ (cid:19) K p ( u ) + (cid:18) p (cid:19) K ( u ) K − p ′ ( u ) , (2.8) ii) L ( u ) ≥ K ( u ) + N ( u ) ≥ N ( u ) , (2.9) iii) L ( u ) ≥ (cid:18) p ′ (cid:19) p K ( u ) + (cid:18) p ′ (cid:19) p − ( K ( u ) + N ( u )) ≥ (cid:18) p ′ (cid:19) p K ( u ) , (2.10)14 or every u ∈ C ∞ Γ − (Ω) .Proof. i) Rising both sides of (2.4) to power 1 p and neglecting N ( u ) ≥ Q p H p − ≥ ps p − Q − ( p − s p H, with H > Q ≥ s ≥ Q = 1 p ( K ( u ) + N ( u ) + ( p − K ( u )) and H = K ( u ) we get L ( u ) ≥ s p − ( K ( u ) + N ( u )) + ( p − s p − (1 − s ) K ( u ) . (2.11)For s = 1 in (2.11) we get (2.9) and neglecting K ( u ) since K ( u ) ≥ s = 1 p ′ = p − p and neglecting K ( u ) ≥ N ( u ) ≥ u in the direction of the unit vector f | f | , on two functions v , w satisfying (2.1) and onadditional term including boundary integral.Since h f, η i ≥ + and |∇ u | p ≥ (cid:12)(cid:12)(cid:12)(cid:12) h f, ∇ u i| f | (cid:12)(cid:12)(cid:12)(cid:12) p , in (2.8)–(2.10) we can replace theirleft-hand sides correspondingly with Z Ω v − p |∇ u | p dx and (cid:18)Z Ω v − p |∇ u | p dx (cid:19) /p . The careful analysis of the proof of Theorem 2.1 shows that (2.4) is an equality if and onlyif (cid:12)(cid:12)(cid:12)(cid:12)Z Ω h f, ∇| u | p i dx (cid:12)(cid:12)(cid:12)(cid:12) = Z Ω h f, ∇| u | p i dx, (2.12)H¨older inequality becomes an equality, i.e., (cid:12)(cid:12)(cid:12)(cid:12) v − p ′ h f, ∇ u i| f | (cid:12)(cid:12)(cid:12)(cid:12) p = k p (cid:12)(cid:12)(cid:12) | v | p ′ | f || u | p − u (cid:12)(cid:12)(cid:12) p ′ , (2.13)for a.e. x ∈ Ω and some constant k > − div f − ( p − v | f | p ′ = w, in Ω . (2.14)However, (2.12) and (2.13) are satisfied if u h f, ∇ u i = | u h f, ∇ u i| , (2.15) h f, ∇ u i = k v | f | p ′ u, k > , (2.16)for a.e. x ∈ Ω. Thus we get the following result for sharpness of Hardy inequality (2.4).15 heorem 2.2.
Suppose p > , n ≥ , Ω is a bounded domain with C smooth boundary ∂ Ω , v > , v − p ∈ L (Ω) and w ≥ for a.e. x ∈ Ω . Then Hardy inequality (2.4) becomesa non-trivial equality if f, v, w, u satisfy (2.14)–(2.16) and u ∈ C ∞ Γ − (Ω) , u . Let us note that the possibility to use a vector function f and two functions v and w in inequalities (2.8)–(2.10) serves for many new Hardy inequalities. We will compare the result in Theorem 2.1 with results in Opic and Kufner [84] andBalinsky et al. [12].
Example 2.1.
Let y ( x ) be a solution of the equation (1.6) with properties listed in Sect.1.1, i.e., v i , w are positive measurable functions, finite a.e. in Ω, so that inequality (1.5)holds, see Opic and Kufner [84], Theorems 14.1 and 14.2. Under these conditions we willprove new Hardy inequalities by means of Theorem 2.1.Suppose that Ω is a bounded domain in R n , n ≥
2. For 1 < p < n we define vectorfunction f = ( f , . . . , f n ) with f i = v i (cid:12)(cid:12)(cid:12)(cid:12) ∂y∂x i (cid:12)(cid:12)(cid:12)(cid:12) p − ∂y∂x i ( | y | p − y ) − , i = 1 , . . . , n, and v ( x ) = inf | ξ | =1 n X i =1 v i ! n X i =1 v i | ξ i | p − ! − p p − . For every u ( x ) ∈ C ∞ (Ω) the following Hardy inequalities hold (cid:18)Z Ω v − p (cid:12)(cid:12)(cid:12)(cid:12) h∇ y, ∇ u i|∇ y | (cid:12)(cid:12)(cid:12)(cid:12) p dx (cid:19) p ≥ p − p Z Ω v | y | p n X i =1 v i (cid:12)(cid:12)(cid:12)(cid:12) ∂y∂x i (cid:12)(cid:12)(cid:12)(cid:12) p − ! p p − | u | p dx p + 1 p Z Ω w | u | p dx Z Ω v | y | p n X i =1 v i (cid:12)(cid:12)(cid:12)(cid:12) ∂y∂x i (cid:12)(cid:12)(cid:12)(cid:12) p − ! p p − | u | p dx − pp , (2.17)and Z Ω v − p (cid:12)(cid:12)(cid:12)(cid:12) h∇ y, ∇ u i|∇ y | (cid:12)(cid:12)(cid:12)(cid:12) p dx ≥ Z Ω w | u | p dx. (2.18)The proof of (2.17) and (2.18) follows from (2.4) and (2.9). Indeed, the vector function f ( x ) satisfies inequality (2.1), i.e., − div f − ( p − v | f | p ′ ≥ w + ( p − n X i =1 v i | y | p (cid:12)(cid:12)(cid:12)(cid:12) ∂y∂x i (cid:12)(cid:12)(cid:12)(cid:12) p − ( p − v | y | p n X i =1 v i (cid:12)(cid:12)(cid:12)(cid:12) ∂y∂x i (cid:12)(cid:12)(cid:12)(cid:12) p − ! p p − ≥ w ( x ) in Ω , n X i =1 v i (cid:12)(cid:12)(cid:12)(cid:12) ∂y∂x i (cid:12)(cid:12)(cid:12)(cid:12) p n X i =1 v i (cid:12)(cid:12)(cid:12)(cid:12) ∂y∂x i (cid:12)(cid:12)(cid:12)(cid:12) p − ! − p p − ≥ inf | ξ | =1 n X i =1 v i | ξ | p n X i =1 v i | ξ | p − ! − p p − = v ( x ) . Since | f | p ′ = n X i =1 v i (cid:12)(cid:12)(cid:12)(cid:12) ∂y∂x i (cid:12)(cid:12)(cid:12)(cid:12) p − ! p p − , we get L ( u ) = Z Ω v − p (cid:12)(cid:12)(cid:12)(cid:12) h f, ∇ u i| f | (cid:12)(cid:12)(cid:12)(cid:12) p dx, K ( u ) = 0 ,K ( u ) = Z Ω v | y | p n X i =1 v i (cid:12)(cid:12)(cid:12)(cid:12) ∂y∂x i (cid:12)(cid:12)(cid:12)(cid:12) p − ! p p − | u | p dx, N ( u ) = Z Ω w | u | p dx, because u = 0 on ∂ Ω. Applying (2.4) and (2.3) we obtain (2.17) and (2.18).
Example 2.2.
Let z be a real-valued function z ∈ W , loc (Ω) in a bounded domain Ω ⊂ R n , ∂ Ω ∈ C , n ≥ z is of one sign a.e. in Ω. Then the following inequalities hold for p >
1, see Davies and Hinz [29] and Balinsky et al. [12], Theorem 1.2.8, Z Ω | ∆ z | p − |h∇ z, ∇ u i| p dx ≥ (cid:18) p (cid:19) p Z Ω | ∆ z || u | p dx (2.19)for every u ∈ C ∞ (Ω).If additional z ∈ C ( ¯Ω) ∩ W , (Ω) then (cid:18)Z Ω | ∆ z | p − |h∇ z, ∇ u i| p dx (cid:19) p ≥ p (cid:18)Z Ω | ∆ z || u | p dx (cid:19) p − p sgn ∆ z Z Γ + h∇ z, η i| u | p dS (cid:18)Z Ω | ∆ z || u | p dx (cid:19) − p − p , (2.20)for every u ∈ C ∞ Γ − (Ω). Here η is the unit outward to Ω normal vector on ∂ Ω, where Γ − , Γ + are defined in (2.2).Inequalities (2.19) and (2.20) follow from (2.8) for f = − (cid:18) p − (cid:19) p − (sgn ∆ z ) ∇ z , v = | ∆ z ||∇ z | − p ′ and w = 0 . Indeed, we get − div f − ( p − v | f | p ′ = (cid:18) p − (cid:19) p − | ∆ z | − (cid:18) p − (cid:19) p − | ∆ z | = 0 , for a.e. in Ω. 17ith the computations L ( u ) = Z Ω v − p (cid:12)(cid:12)(cid:12)(cid:12) h f, ∇ u i| f | (cid:12)(cid:12)(cid:12)(cid:12) p dx = Z Ω | ∆ z | p − |h∇ z, ∇ u i| p dx,K ( u ) = − Z Γ + h f, η i| u | p dS = (cid:18) p − (cid:19) p − (sgn ∆ z ) Z Γ + h∇ z, η i| u | p dS,K ( u ) = Z Ω v | f | p ′ | u | p dx = (cid:18) p − (cid:19) p Z Ω | ∆ z || u | p dx,N ( u ) = 0 . applying (2.8) and (2.10) we obtain (2.19) and (2.20), respectively. We illustrate below the possibility to choose a vector function f in order to obtain sharpHardy inequality. Inequality with weight the first eigenfunction of the p-Laplacian
Let ϕ be the first eigenfunction of the p-Laplacian in a bounded domain Ω ⊂ R n , p > n ≥ λ > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ∆ p ϕ = λ | ϕ | p − ϕ, in Ω ,ϕ | ∂ Ω = 0 . Let us define the vector function f = |∇ ϕ | p − ∇ ϕ | ϕ | p − ϕ in every domain Ω , ¯Ω ⊂ Ω such thatfor f = f . . . f n we have f i ∈ C (Ω ) and − div f = − ∆ p ϕ | ϕ | p − ϕ + ( p − |∇ ϕ | p | ϕ | p = λ + ( p − | f | p ′ x ∈ Ω , i.e., v = 1 and ω = λ in (2.1).If we fix u ∈ C ∞ (Ω), then supp u ⊂ Ω so that we can apply Theorem 2.1 and obtainthe inequality L ( u ) ≥ (cid:18) p (cid:19) p [( p − K ( u ) + N ( u )] p K p − ( u ) , (2.21)where L ( u ) = Z Ω (cid:12)(cid:12)(cid:12)(cid:12) h∇ ϕ, ∇ u i|∇ ϕ | (cid:12)(cid:12)(cid:12)(cid:12) p dx, K ( u ) = Z Ω (cid:12)(cid:12)(cid:12)(cid:12) ∇ ϕϕ (cid:12)(cid:12)(cid:12)(cid:12) p | u | p dx, N ( u ) = λ Z Ω | u | p dx. Note that under arguments of completeness the inequality (2.21) holds for every u ∈ W ,p (Ω), moreover simple computation gives us that inequality (2.21) is sharp, i.e., be-comes an equality for u ( x ) = ϕ ( x ).We consider the case p = 2, n = 3 and Ω = B . In this case the first eigenfunction is18 = √ πrπr , r = | x | , and the eigenvalue is λ = π , see Vladimirov [96], Sect. 28.1.Now we have f = ∇ ϕϕ = (cid:18) π cot πr − r (cid:19) xr , | f | = (cid:18) π cot πr − r (cid:19) and − div f = | f | + π .Applying Theorem 2.1 to L ( u ) = Z B |∇ u | dx , K ( u ) = Z B | f | u dx and N ( u ) = π Z B u dx we get L ( u ) ≥ (cid:0) K ( u ) + 2 N ( u ) + N ( u ) K − ( u ) (cid:1) for u ∈ C ∞ ( B ) . (2.22)Using the series expansion for the function cot( z ), see Remmert [91], we obtain (cid:18) π cot πr − r (cid:19) = ∞ X k =1 rr − k ! = 4 r ( r − " ∞ X k =2 r − r − k = r ( r − " r + 1 + 2 ∞ X k =2 r − r − k , (2.23)for the kernel of K ( u ) and r ∈ (0 , Z B |∇ u | dx ≥ Z B u (1 − | x | ) | x | + 1 + 2 ∞ X k =2 | x | − | x | − k ! dx + 2 π Z B u dx + π (cid:16)R B u dx (cid:17) R B u (1 −| x | ) (cid:16) | x | +1 + 2 P ∞ k =2 | x |− | x | − k (cid:17) dx . (2.24)Since the last term in (2.24) is positive, 2 | x | + 1 ≥ − | x | = d ( x ) = dist( x, ∂B ), wecan rewrite (2.24) as Z B |∇ u | dx ≥ Z B u d ( x ) dx + A ( u ) u ∈ W ,p ( B ) , (2.25)where A ( u ) = 14 Z B − | x | | x | + 2 ∞ X k =2 | x | − | x | − k ! − u dx + 12 π Z B u dx + 14 π (cid:16)R B u dx (cid:17) R B u (1 −| x | ) (cid:16) | x | +1 + 2 P ∞ k =2 | x |− | x | − k (cid:17) dx > . Inequality (2.25) has an optimal constant 14 and moreover it is sharp, i.e., for function u ( x ) = √ π | x | π | x | it becomes an equality. The above example shows that sharp inequality191.11) with optimal constant 14 , see Definition 1.1, is possible but for more complicatedadditional term A ( u ). Thus, in this special case we give a positive answer to the questionof Brezis and Marcus [25]. Hardy inequalities in an annulus and in a ball
Let us define for p > p ′ = pp − n ≥ m = p − np − ≤ r < R the sets of functions: M ( r, R ) = u : Z B R \ B r (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx < ∞ , ≤ r < R, and (cid:12)(cid:12)(cid:12) R m − ˆ R m (cid:12)(cid:12)(cid:12) − p Z ∂B ˆ R | u | p dS → , ˆ R → R − , m = 0 , (cid:12)(cid:12)(cid:12)(cid:12) ln R ˆ R (cid:12)(cid:12)(cid:12)(cid:12) − p Z ∂B ˆ R | u | p dS → , ˆ R → R − , m = 0 . (2.26) Proposition 2.1.
For u ∈ M ( r, R ) and < r < R the following Hardy inequalities hold: • for m = 0 , i.e., p = n Z B R \ B r (cid:12)(cid:12)(cid:12)(cid:12) < x, ∇ u > | x | (cid:12)(cid:12)(cid:12)(cid:12) p dx ! p ≥ (cid:12)(cid:12)(cid:12)(cid:12) n − pp (cid:12)(cid:12)(cid:12)(cid:12) Z B R \ B r | u | p | x | ( n − p ′ | R m − | x | m | p dx ! p + 1 p r − n | R m − r m | − p Z ∂B r | u | p dS Z B R \ B r | u | p | x | ( n − p ′ | R m − | x | m | p dx ! − p ′ . (2.27) For function u k ( x ) = (cid:18) R m − | x | m m (cid:19) k , k > p ′ inequality (2.27) becomes an equality; • for m = 0 , i. e., p = n Z B R \ B r (cid:12)(cid:12)(cid:12)(cid:12) < x, ∇ u > | x | (cid:12)(cid:12)(cid:12)(cid:12) n dx ! n ≥ n − n Z B R \ B r | u | n | x | n (cid:12)(cid:12)(cid:12)(cid:12) ln R | x | (cid:12)(cid:12)(cid:12)(cid:12) n dx n + 1 n (cid:18) r ln Rr (cid:19) − n Z ∂B r | u | n dS Z B R \ B r | u | n | x | n (cid:12)(cid:12)(cid:12)(cid:12) ln R | x | (cid:12)(cid:12)(cid:12)(cid:12) n dx − n ′ . (2.28)20 or function u s ( x ) = (cid:18) ln R | x | (cid:19) s , s > n ′ inequality (2.28) becomes an equality.Proof. Let the function ψ ( x ) be a solution of the problem: ∆ p ψ = 0 , in B R \ B r ,ψ | ∂B R = 0 , ψ | ∂B r = 1 , , then ψ ( x ) = R m − | x | m R m − r m , m = 0 , ln R | x | ln Rr , m = 0 . (2.29)Indeed • for m = 0 we have ∇ ψ = − m | x | m − x | x | R m − r m , |∇ ψ | p − ∇ ψ = −| m | p − m | x | ( m − p − x | x | (cid:18) R m − r m (cid:19) p − , ∆ p ψ = div (cid:0) |∇ ψ | p − ∇ ψ (cid:1) = −| m | p − m | x | ( m − p − − (cid:18) R m − r m (cid:19) p − [( m − p −
1) + n −
1] = 0 , because ( m − p −
1) + n − (cid:18) p − np − − (cid:19) ( p −
1) + n − • for m = 0 we have ∇ ψ = − | x | Rr x | x | , |∇ ψ | n − ∇ ψ = − | x | n − (cid:0) ln Rr (cid:1) n − x | x | , ∆ n ψ = div (cid:0) |∇ ψ | n − ∇ ψ (cid:1) = −| x | − n (cid:18) ln Rr (cid:19) n − [ − ( n −
1) + n −
1] = 0 . Using the function ψ in (2.29) we define the vector function f ( x ) in B R \ B r as f = |∇ ψ | p − | ψ | p − ∇ ψψ and let us check that f ( x ) = −| x | − n x (cid:18) R m − | x | m m (cid:19) − p , m = 0 , −| x | − n x (cid:18) ln R | x | (cid:19) − n , m = 0 . Indeed 21 for m = 0 we have f = −| m | p − m | x | ( m − p − − x ( R m − | x | m ) − | R m − | x | m | − p = −| x | − n x (cid:18) R m − | x | m m (cid:19) − p , because ( m − p − − (cid:18) p − np − − (cid:19) ( p − − − n ; • for m = 0 we have f = −| x | − ( n − (cid:18) ln R | x | (cid:19) − n x | x | = −| x | − n x (cid:18) ln R | x | (cid:19) − n . Note that the outward normal η to B ˆ R \ B r , r < ˆ R < R is defined as η | ∂B ˆ R = x | x | | ∂B ˆ R , η | ∂B r = − x | x | | ∂B r . Moreover, we get for u ∈ M ( r, R ) and • for m = 0 | f | p ′ = | x | (1 − n ) p ′ (cid:18) R m − | x | m m (cid:19) − p , Z ∂ ( B ˆ R \ B r ) h f, η i| u | p dS ≥ , and − div f = − div (cid:12)(cid:12)(cid:12)(cid:12) ∇ ψψ (cid:12)(cid:12)(cid:12)(cid:12) p − ∇ ψψ ! = − ∆ p ψψ p − + | x | − n * x, ∇ (cid:18) R m − | x | m m (cid:19) − p + = ( p − | x | − n (cid:18) R m − | x | m m (cid:19) − p | x | m = ( p − | x | m − n (cid:18) R m − | x | m m (cid:19) − p = ( p − | f | p ′ ; (2.30) • for m = 0 | f | n ′ = | x | − n (cid:18) ln R | x | (cid:19) − n , Z ∂ ( B R \ B r ) h f, η i| u | n dS ≥ , − div f = − div (cid:12)(cid:12)(cid:12)(cid:12) ∇ ψψ (cid:12)(cid:12)(cid:12)(cid:12) n − ∇ ψψ ! = − ∆ n ψψ n − + | x | − n * x, ∇ (cid:18) ln R | x | (cid:19) − n + = − (1 − n ) | x | − n (cid:12)(cid:12)(cid:12)(cid:12) ln R | x | (cid:12)(cid:12)(cid:12)(cid:12) − n (cid:28) x, x | x | (cid:29) = ( n − | x | − n (cid:18) ln R | x | (cid:19) − n = ( n − | f | n ′ . (2.31)Since the vector function f ( x ) satisfies (2.1) with v = 1 , w ≡ B ˆ R \ ¯ B r we obtain inequalities (2.27), (2.28) after the limitˆ R → R .We will prove the sharpness of (2.27) only for m > m < k > p ′ the integral I m = Z B R \ B r dx | x | ( n − p ′ ( R m − | x | m ) p (1 − k ) . With a change of variables y = xR and ρ = | y | we get I m = R m (1 − p + kp ) Z B R \ B r dy | y | ( n − p ′ (1 − | y | m ) p (1 − k ) = R m (1 − p + kp ) ω n Z r/R ρ m − dρ (1 − ρ m ) p (1 − k ) = ω n m − ( R m − r m ) − p + kp (1 − p + kp ) − , (2.32)where we use ( n − p ′ = n − m and 1 − p + kp > k > p ′ = p − p .23ith u k ( x ) = (cid:18) R m − | x | m m (cid:19) k for the left-hand side of (2.27) we get( lhs ) = Z B R \ B r (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u k i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx ! p = k Z B R \ B r | x | ( m − p (cid:18) R m − | x | m m (cid:19) ( k − p dx ! p = km − k Z B R \ B r dx | x | ( n − p ′ ( R m − | x | m ) p (1 − k ) ! p = km − k I p m = kω p n m − − p + kpp ( R m − r m ) − p + kpp (1 − p + kp ) − p , where we use ( m − p = (cid:18) p − np − − (cid:19) p = − ( n − p ′ .For the terms in the right-hand side of (2.27) using the expression (2.32) for I m we get( rhs ) = p − np Z B R \ B r | u k | p dx | x | ( n − p ′ ( R m − | x | m ) p ! p = p − np m − k Z B R \ B r dx | x | ( n − p ′ ( R m − | x | m ) p (1 − k ) ! p = p − np m − m − k +1 I p m = p − p m − k +1 I p m = p − p ω p n m − − p + kpp ( R m − r m ) − p + kpp (1 − p + kp ) − p . (2.33)( rhs ) = 1 p r − n ( R m − r m ) − p Z ∂B r | u k | p dS × Z B R \ B r | u k | p dx | x | ( n − p ′ ( R m − | x | m ) p ! − p ′ = 1 p ω n r n − r − n ( R m − r m ) − p m − kp ( R m − r m ) pk (cid:16) m − pk I m (cid:17) − p ′ = 1 p ω p n m − − p + kpp ( R m − r m ) − p + kpp (1 − p + kp ) p ′ . (2.34)24dding (2.33) and (2.34) we obtain( rhs ) + ( rhs ) = 1 p ω p n m − − p + kpp ( R m − r m ) − p + kpp × " p − − p + kp ) p + (1 − p + kp ) p ′ = kω p n m − − p + kpp ( R m − r m ) − p + kpp (1 − p + kp ) − p = Z B R \ B r (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx ! p = ( lhs ) , which proves the sharpness of (2.27) for m > I = Z B R \ B r | y | − n (cid:18) ln 1 | y | (cid:19) n ( s − dy = ω n Z r/R (cid:18) ln 1 ρ (cid:19) n ( s − ρ − dρ = ω n (cid:18) ln Rr (cid:19) − n + ns (1 − n + ns ) − , where we use 1 − n + ns > s > n ′ .With u s ( x ) = (cid:18) ln R | x | (cid:19) s for the left-hand side of (2.28) we get( lhs ) = Z B R \ B r (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u s i| x | (cid:12)(cid:12)(cid:12)(cid:12) n dx ! n = s Z B R \ B r | x | − n (cid:18) ln R | x | (cid:19) ( s − n dx ! n = sI n = sω n n (cid:18) ln Rr (cid:19) − n + snn (1 − n + sn ) − n . For the terms in the right-hand side we get( rhs ) = n − n Z B R \ B r | u s | n dx | x | n (cid:16) ln R | x | (cid:17) n n = n − n Z B R \ B r | x | − n (cid:18) ln R | x | (cid:19) n ( s − dx ! n = n − n I n = n − n ω n n (1 − n + sn ) − n (cid:18) ln Rr (cid:19) − n + snn , (2.35)25nd ( rhs ) = 1 n (cid:18) r ln Rr (cid:19) − n Z ∂B r | u s | n dS Z B R \ B r | u s | n dx | x | n (cid:16) ln R | x | (cid:17) n − n ′ = 1 n ω n r − n (cid:18) ln Rr (cid:19) − n + sn Z B R \ B r dx | x | n (cid:16) ln R | x | (cid:17) n ( s − − n ′ = 1 n ω n (cid:18) ln Rr (cid:19) − n + sn I n ′ = 1 n ω n n (cid:18) ln Rr (cid:19) − n + sn (cid:18) ln Rr (cid:19) − (1 − n + sn ) n ′ (1 − n + sn ) n ′ = 1 n ω n n (cid:18) ln Rr (cid:19) n (1 − n + sn ) n − n . (2.36)Adding (2.35) and (2.36) we obtain( rhs ) + ( rhs ) = ω n n (cid:18) ln Rr (cid:19) − n + snn " − n + sn ) n n − n + (1 − n + sn ) n − n n = sω n n (cid:18) ln Rr (cid:19) − n + snn (1 − n + sn ) − n = Z B R \ B r (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u s i| x | (cid:12)(cid:12)(cid:12)(cid:12) n ! n = ( lhs ) , which proves the sharpness of (2.28).Using Proposition 2.1 we will obtain Hardy inequalities in a ball which are sharp for p > n and optimal for p > Proposition 2.2.
For functions u ∈ M (0 , R ) defined in (2.26) the following inequalitieshold: ) for m > , i.e., p > n (cid:18)Z B R (cid:12)(cid:12)(cid:12)(cid:12) < x, ∇ u > | x | (cid:12)(cid:12)(cid:12)(cid:12) p dx (cid:19) p ≥ p − np (cid:18)Z B R | u | p | x | ( n − p ′ | R m − | x | m | p dx (cid:19) p + 1 p R n − p limsup r → (cid:20) r − n Z ∂B r | u | p dS (cid:21) × (cid:18)Z B R | u | p | x | ( n − p ′ | R m − | x | m | p dx (cid:19) − p ′ . (2.37) For the functions u k ( x ) = (cid:18) R m − | x | m m (cid:19) k , k > p ′ , inequality (2.37) becomes an equalityand the constant p − np in (2.37) is optimal.ii) for m < , i.e., p < n (cid:18)Z B R (cid:12)(cid:12)(cid:12)(cid:12) < x, ∇ u > | x | (cid:12)(cid:12)(cid:12)(cid:12) p dx (cid:19) p ≥ (cid:12)(cid:12)(cid:12)(cid:12) p − np (cid:12)(cid:12)(cid:12)(cid:12) (cid:18)Z B R | u | p | x | ( n − p ′ | R m − | x | m | p dx (cid:19) p + 1 p R n − p limsup r → (cid:20) r − p Z ∂B r | u | p dS (cid:21) × (cid:18)Z B R | u | p | x | ( n − p ′ | R m − | x | m | p dx (cid:19) − p ′ . (2.38) The constant (cid:12)(cid:12)(cid:12)(cid:12) p − np (cid:12)(cid:12)(cid:12)(cid:12) in (2.38) is optimal.iii) for m = 0 , i.e., p = n (cid:18)Z B R (cid:12)(cid:12)(cid:12)(cid:12) < x, ∇ u > | x | (cid:12)(cid:12)(cid:12)(cid:12) n dx (cid:19) n ≥ n − n Z B R | u | n | x | n (cid:12)(cid:12)(cid:12)(cid:12) ln R | x | (cid:12)(cid:12)(cid:12)(cid:12) n dx n + 1 n limsup r → "(cid:18) r ln Rr (cid:19) − n Z ∂B r | u | n dS B R | u | n | x | n (cid:12)(cid:12)(cid:12)(cid:12) ln R | x | (cid:12)(cid:12)(cid:12)(cid:12) n dx − n ′ . (2.39)27 he constant n − n in (2.39) is optimal.Proof. Let us apply (2.27) and (2.28) in Proposition 2.1. Then after the limit r → u ∈ M (0 , R ). More precisely,i) Inequality (2.37) becomes equality for u k ( x ) = (cid:18) R m − | x | m m (cid:19) k , k > p ′ andhence the constant p − nn is optimal. Indeed, the sharpness of (2.37) is a consequence ofthe sharpness of (2.27) and the limit r → u k ∈ M (0 , R ) for k > p ′ .ii) Note that for m < r → p r − n | R m − r m | − p Z ∂B r | u | p dS = limsup r → p r − n + m (1 − p ) | R m r − m − | − p Z ∂B r | u | p dS = limsup r → p r − p | R m r − m − | − p Z ∂B r | u | p dS = limsup r → p r − p Z ∂B r | u | p dS. We will prove that the constant (cid:12)(cid:12)(cid:12)(cid:12) p − np (cid:12)(cid:12)(cid:12)(cid:12) in (2.38) is optimal using the function u ε ( x ) = | x | − | m | p ′ (1 − ε ) (cid:16) R | m | − | x | | m | (cid:17) p ′ (1+ ε ) for 0 < ε <
1. Note that u ε ( x ) ∈ M (0 , R ) because forthe power of | x | we have −| m | p ′ (1 − ε ) = ε | m | p ′ + p − np = ε | m | p ′ + n ( p − p + 1 − n > − n, hence u ε ( x ) is integrable at 0.Ignoring the boundary term in (2.38) and rising both sides to p -th power for the left-hand side we obtain( lhs ) = Z B R (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u ε i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx = Z B R | x | −| m | ( p − − ε ) − p (cid:16) R | m | − | x | | m | (cid:17) ( p − ε ) − p × (cid:12)(cid:12)(cid:12)(cid:12) | m | p ′ [(1 − ε ) R | m | + 2 ε | x | | m | ] (cid:12)(cid:12)(cid:12)(cid:12) p dx ≤ (cid:12)(cid:12)(cid:12)(cid:12) p − np (cid:12)(cid:12)(cid:12)(cid:12) p (1 + ε ) p R | m | p Z B R | x | ( p − n )(1 − ε ) − p (cid:16) R | m | − | x | | m | (cid:17) ( p − ε ) − p dx. rhs )= (cid:12)(cid:12)(cid:12)(cid:12) p − np (cid:12)(cid:12)(cid:12)(cid:12) p Z B R | u ε | p | x | ( n − p ′ | R m − | x | m | p dx = (cid:12)(cid:12)(cid:12)(cid:12) p − np (cid:12)(cid:12)(cid:12)(cid:12) p Z B R | x | − | m | p ′ (1 − ε ) p (cid:0) R | m | − | x | | m | (cid:1) pp ′ (1+ ε ) − p R | m | p | x | | m | p | x | ( n − p ′ dx = (cid:12)(cid:12)(cid:12)(cid:12) p − np (cid:12)(cid:12)(cid:12)(cid:12) p R | m | p Z B R | x | −| m | ( p − − ε ) − p (cid:16) R | m | − | x | | m | (cid:17) ( p − ε ) − p dx, and hence 1 < ( lhs )( rhs ) < (1 + ε ) p because (cid:18) | m | p ′ (cid:19) p = (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) p − np − (cid:12)(cid:12)(cid:12)(cid:12) p − p (cid:19) p = (cid:12)(cid:12)(cid:12)(cid:12) p − np (cid:12)(cid:12)(cid:12)(cid:12) p . For ε → (cid:12)(cid:12)(cid:12)(cid:12) p − np (cid:12)(cid:12)(cid:12)(cid:12) p is optimal.(iii) We will prove that the constant n − n in (2.39) is optimal using the function u s ( x ) = (cid:18) ln R | x | (cid:19) s , for r < | x | < R, (cid:18) ln Rr (cid:19) s , for 0 ≤ | x | ≤ r , with s = n − n (1 + ε ), 0 < ε .Ignoring the boundary term in (2.39) and rising both sides to n -th power we obtainthe inequality Z B R (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u s i| x | (cid:12)(cid:12)(cid:12)(cid:12) n dx ≥ (cid:18) n − n (cid:19) n Z B R | u s | n | x | n (cid:12)(cid:12)(cid:12) ln R | x | (cid:12)(cid:12)(cid:12) n dx. (2.40)For the left-hand side and for the right-hand side of (2.40) as in Proposition 2.1 we get( lhs ) = Z B R (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u s i| x | (cid:12)(cid:12)(cid:12)(cid:12) n dx = s n Z B R \ B r | x | − n (cid:18) ln R | x | (cid:19) n ( s − dx = s n ω n Z r /R ρ − (cid:18) ln 1 ρ (cid:19) n ( s − = s n ω n (cid:18) ln Rr (cid:19) − n + sn (1 − n + sn ) − . rhs ) = (cid:18) n − n (cid:19) n Z B R | u s | n | x | n (cid:12)(cid:12)(cid:12) ln R | x | (cid:12)(cid:12)(cid:12) n dx = (cid:18) n − n (cid:19) n Z B R \ B r | x | − n (cid:18) ln R | x | (cid:19) n ( s − dx + (cid:18) n − n (cid:19) n (cid:18) ln Rr (cid:19) sn Z B r | x | − n (cid:18) ln R | x | (cid:19) n dx = (cid:18) n − n (cid:19) n ω n (cid:18) ln Rr (cid:19) − n + sn (1 − n + sn ) − + (cid:18) n − n (cid:19) n (cid:18) ln Rr (cid:19) sn ω n Z r /R ρ − (cid:18) ln 1 ρ (cid:19) − n dρ = (cid:18) n − n (cid:19) n ω n (cid:18) ln Rr (cid:19) − n + sn (cid:20) − n + sn + 1 n − (cid:21) = (cid:18) n − n (cid:19) n ω n (cid:18) ln Rr (cid:19) − n + sn sn (1 − n + sn )( n − . Since s = (1 + ε ) n − n > n ′ , then 1 ≤ ( lhs )( rhs ) = (1 + ε ) n − and the sharpness of (2.40) isproved. The aim of this section is to prove Hardy inequality in a ball B R centered at zero withradius 0 < R < ∞ , B R ⊂ R n , n ≥ ∂B R and at the origin and with an additional logarithmic term. We generalize the results inBarbatis et al. [13] where the weights are singular only on the boundary of the domain.Let p > p ′ = pp − n ≥ m = p − np − Lemma 2.1.
For every p ≥ , there exists a = a ( p ) < and τ > such that for every τ > τ the function Z ( s ) = (cid:18) p ′ (cid:19) p − (cid:18) −
11 + ln τ − s + a (1 + ln τ − s ) (cid:19) , satisfies Z ( s ) ∈ C ( −∞ , , Z > , Z ′ < , Z ( −∞ ) = (cid:18) p ′ (cid:19) p − , (2.41) and is a solution of the inequality − Z ′ + ( p − Z − ( p − Z p ′ ≥ H ( s ) , (2.42) where H ( s ) = (cid:18) p ′ (cid:19) p (cid:18) p p −
1) 1(1 + ln τ − s ) (cid:19) . roof. Let us denote for simplicity y ( s ) = 11 + ln τ − s , so that Z ( s ) = (cid:18) p ′ (cid:19) p − (1 − y + ay ) , Z ′ ( s ) = (cid:18) p ′ (cid:19) p − ( − y + 2 ay )Expanding Z p ′ ( y ) for a small y near y = 0 in a Taylor polynomial up to third order weobtain Z p ′ = (cid:18) p ′ (cid:19) p (cid:26) − pp − y + pp − (cid:18) a + 1 p − (cid:19) y pp − (cid:20) − ap − − p − p − (cid:21) y o ( y ) (cid:27) . Then if a < − p − p −
1) we get − Z ′ + ( p − Z − ( p − Z p ′ = (cid:18) p ′ (cid:19) p (cid:20) p p − y + pp − (cid:18) − a + p − p − (cid:19) y + o ( y ) (cid:21) ≥ (cid:18) p ′ (cid:19) p (cid:18) p p − y (cid:19) . With this choice of a inequalities (2.42) and Z ′ ( s ) < τ such that Z ( s ) >
0, i.e., 1 − y − | a | y > < y < y = 1 − p | a |− | a | . (2.43)Let τ = e y − then for every τ > τ we get Z ( s ) > f , f = (cid:12)(cid:12)(cid:12)(cid:12) ∇ ψψ (cid:12)(cid:12)(cid:12)(cid:12) p − ∇ ψψ Z (ln ψ ) in B R \ B r , (2.44)where ψ ( x ) is defined in (2.29) and Z is given in Lemma 2.1. Proposition 2.3.
The vector function f = { f , . . . , f n } in (2.44) satisfies f j ∈ C ( B R \ B r ) and − div f − ( p − | f | p ′ ≥ w, in B R \ B r , (2.45) where w = (cid:12)(cid:12)(cid:12)(cid:12) ∇ ψψ (cid:12)(cid:12)(cid:12)(cid:12) p H (ln ψ ) and H ( s ) is defined in Lemma 2.1.Moreover, for every u ∈ W ,p ( B R ) , the following inequality holds L ( u ) ≥ N ( u ) , (2.46) where L ( u ) = Z B R \ B r (cid:12)(cid:12)(cid:12)(cid:12) h f, ∇ u i| f | (cid:12)(cid:12)(cid:12)(cid:12) p dx, and for m = 0 , i.e., p = nN ( u ) = Z B R \ B r w | u | p dx = (cid:12)(cid:12)(cid:12)(cid:12) p − np (cid:12)(cid:12)(cid:12)(cid:12) p Z B R \ B r " p p −
1) 1ln ψeτ | u | p | x | ( n − p ′ | R m − | x | m | p dx (2.47) • for m = 0 , i.e., p = nN ( u ) = Z B R \ B r w | u | n dx = (cid:18) n − n (cid:19) n Z B R \ B r " n n −
1) 1ln ψeτ | u | n | x | n (cid:12)(cid:12)(cid:12) ln R | x | (cid:12)(cid:12)(cid:12) n dx, (2.48) where τ > τ = e y − and y is defined in (2.43).Proof. Let us check that the function f satisfies (2.45). Indeed, − div f = − (cid:18) ∆ p ψ | ψ | p − − ( p − (cid:12)(cid:12)(cid:12)(cid:12) ∇ ψψ (cid:12)(cid:12)(cid:12)(cid:12) p (cid:19) Z (ln ψ ) − (cid:12)(cid:12)(cid:12)(cid:12) ∇ ψψ (cid:12)(cid:12)(cid:12)(cid:12) p Z ′ (ln ψ )= (cid:12)(cid:12)(cid:12)(cid:12) ∇ ψψ (cid:12)(cid:12)(cid:12)(cid:12) p h − Z ′ + ( p − Z − ( p − Z p ′ i + ( p − (cid:12)(cid:12)(cid:12)(cid:12) ∇ ψψ (cid:12)(cid:12)(cid:12)(cid:12) p Z p ′ ≥ ( p − | f | p ′ + (cid:12)(cid:12)(cid:12)(cid:12) ∇ ψψ (cid:12)(cid:12)(cid:12)(cid:12) p H (ln ψ ) . Since f as a function of x has the form f ( x ) = − x | x | − n (cid:18) R m − | x | m m (cid:19) − p Z (ln ψ ) , m = 0 , − x | x | − n (cid:18) ln R | x | (cid:19) − n Z (ln ψ ) , m = 0 , then h f, η i | ∂B r >
0. Ignoring the boundary term over ∂B r we can apply (2.9) in thedomain B ˆ R \ B r , r < ˆ R < R for v = 1 and w = (cid:12)(cid:12)(cid:12)(cid:12) ∇ ψψ (cid:12)(cid:12)(cid:12)(cid:12) p H (ln ψ ) to obtain (2.46) after thelimit ˆ R → R .The inequality with additional logarithmic weight in a ball can be obtained only for p > n , i.e., m >
0. In the case m ≤ ψ ( x ) defined in (2.29) satisfies ψ ( x ) = r | m | ( R | m | − | x | | m | ) | x | | m | ( R | m | − r | m | ) → , and ln R | x | / ln Rr → r → , N ( u ) defined in (2.47), (2.48) are the same as the inequalities(2.38) and (2.39) without boundary and logarithmic terms. When m >
0, i.e., p > n , thefunction ψ ( x ) = R m − | x | m R m − r m → R m − | x | m R m for r → N ( u ). Proposition 2.4.
For m > , i.e., p > n , the following inequality holds for every u ∈ W ,p ( B R ) Z B R |∇ u | p dx ≥ Z B R (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx ≥ (cid:18) p − np (cid:19) p Z B R " p p −
1) 1ln R m −| x | m eτ R m | u | p | x | ( n − p ′ | R m − | x | m | p dx, (2.49) where τ = e y − , y is defined in (2.43) with a = − p − p − .Proof. Since the function ψ ( x ) = R m − | x | m R m − r m → R m − | x | m R m for r → r → N ( u ) from the expression (2.47) we obtain (2.49). By continuity we get τ = τ in (2.47) For n ≥ p > B R − div( |∇ φ | p − ∇ φ ) = w ( | x | ) in B R ,φ = 0 on ∂B R , (2.50)where Z R s n − w ( s ) ds < ∞ . (2.51)We choose a function w ( | x | ) such that the function φ has a simple form.For this purpose let us apply the result in Biezuner et al. [23], where it is shown thatthe solution of (2.50), (2.51) is given by φ ( | x | ) = Z R | x | θ − np − (cid:18)Z θ s n − w ( s ) ds (cid:19) p − dθ. (2.52)Indeed, from the invariance of (2.50) under rotation, problem (2.50) is equivalent tothe boundary value problem for ordinary differential equation − ( r n − | φ ′ | p − φ ′ ) ′ = r n − w ( r ) , < r < R,φ ( R ) = 0 , φ ′ (0) = 0 . (2.53)Integrating twice the equation in (2.53) and applying boundary conditions we obtain(2.52). 33et us chose the function w ( | x | ) = | x | − δ , δ ∈ (0 , n ), then (2.51) holds and from (2.52)we have φ ( | x | ) = p − p − δ ( n − δ ) − p − (cid:16) R p − δp − − | x | p − δp − (cid:17) for δ = p, ( n − p ) − p − ln R | x | for δ = p. For ε ∈ (0 , R ) we define the vector function f = |∇ φ | p − ∇ φ | φ | p − φ . Then f ∈ C ( B R \ B ε ), ε < ˆ R < R and f satisfies the equation − div f = − ∆ p φ | φ | p − φ + ( p − |∇ φ | p | φ | p = w ( | x | ) | φ | p − + ( p − | f | pp − in B R \ B ε . According to Corollary 2.1, the following Hardy inequality holds for u ∈ C ∞ ( B R ), withsupp u ⊂ B ˆ R L ε ( u ) ≥ (cid:18) p (cid:19) p | ( p − K ε ( u ) + K ε ( u ) + N ε ( u ) | p K p − ε ( u ) , u ∈ C ∞ ( B R ) , (2.54)where K ε ( u ) = Z S ε h f, η i| u | p dS + Z ∂B ˆ R h f, η i| u | p dS = Z S ε h f, η i| u | p dS. Here η is the unit outward normal vector to ∂ ( B ˆ R \ B ε ). The expressions for L ε ( u ), K ε ( u )and N ε ( u ) in (2.54) are correspondingly: • in the case δ = p Z B R \ B ε |∇ u | p dx ≥ L ε ( u ) = Z B ˆ R \ B ε (cid:12)(cid:12)(cid:12)(cid:12) h∇ φ, ∇ u i|∇ φ | (cid:12)(cid:12)(cid:12)(cid:12) p dx = Z B R \ B ε (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx,K ε ( u ) = Z B ˆ R \ B ε (cid:12)(cid:12)(cid:12)(cid:12) ∇ φφ (cid:12)(cid:12)(cid:12)(cid:12) p | u | p dx = (cid:12)(cid:12)(cid:12)(cid:12) p − δp − (cid:12)(cid:12)(cid:12)(cid:12) p Z B R \ B ε | u | p | x | ( δ − pp − (cid:12)(cid:12)(cid:12) R p − δp − − | x | p − δp − (cid:12)(cid:12)(cid:12) p dx,N ε ( u ) = Z B ˆ R \ B ε ) w ( | x | ) | φ | p − | u | p dx = ( n − δ ) (cid:12)(cid:12)(cid:12)(cid:12) p − δp − (cid:12)(cid:12)(cid:12)(cid:12) p − Z B R \ B ε | u | p | x | δ (cid:12)(cid:12)(cid:12) R p − δp − − | x | p − δp − (cid:12)(cid:12)(cid:12) p − dx, (2.55) • in the case δ = pK ε ( u ) = Z B R \ B ε | u | p | x | p (cid:12)(cid:12)(cid:12) ln R | x | (cid:12)(cid:12)(cid:12) p dx, N ε ( u ) = | n − p | Z B R \ B ε | u | p | x | p (cid:12)(cid:12)(cid:12) ln R | x | (cid:12)(cid:12)(cid:12) p − dx. (2.56)34ince ∇ φ = − ( n − δ ) − p − | x | − δp − x | x | , η | S ε = − xε , and h∇ φ, η i = ( n − δ ) − p − | ε | − δp − ≥ , for x ∈ S ε , we get Z S ε h f, η i| u | p dS = ( n − δ ) − p − ε − δp − Z S ε | u | p dS ≥ . Hence, neglecting K ε (2.54) becomes L ε ( u ) ≥ (cid:18) p (cid:19) p [( p − K ε ( u ) + N ε ( u )] p K p − ε ( u ) , u ∈ C ∞ ( B R ) . After the limit ε → L ( u ) ≥ (cid:18) p (cid:19) p [( p − K ( u ) + N ( u )] p K p − ( u ) , u ∈ C ∞ ( B R ) . (2.57)holds in B R with L ( u ), K ( u ) and N ( u ) defined in (2.55) and (2.56) for ε = 0, i.e., • in the case δ = pL ( u ) = Z B R (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx,K ( u ) = (cid:12)(cid:12)(cid:12)(cid:12) p − δp − (cid:12)(cid:12)(cid:12)(cid:12) p Z B R | u | p | x | ( δ − pp − (cid:12)(cid:12)(cid:12) R p − δp − − | x | p − δp − (cid:12)(cid:12)(cid:12) p dx,N ( u ) = ( n − δ ) (cid:12)(cid:12)(cid:12)(cid:12) p − δp − (cid:12)(cid:12)(cid:12)(cid:12) p − Z B R | u | p | x | δ (cid:12)(cid:12)(cid:12) R p − δp − − | x | p − δp − (cid:12)(cid:12)(cid:12) p − dx ; (2.58) • in the case δ = pK ( u ) = Z B R | u | p | x | p (cid:12)(cid:12)(cid:12) ln R | x | (cid:12)(cid:12)(cid:12) p dx, N ( u ) = | n − p | Z B R | u | p | x | p (cid:12)(cid:12)(cid:12) ln R | x | (cid:12)(cid:12)(cid:12) p − dx. (2.59)From (2.11) and (2.57) we get L ( u ) ≥ ( p − s p − (1 − s ) K ( u ) + s p − N ( u ) . (2.60)In particular, for s = p − p in (2.60) a ‘linear’ form of Hardy inequality holds. Lemma 2.2.
The inequalities K ( u ) < ∞ and N ( u ) < ∞ for K ( u ) and N ( u ) defined in(2.58) and (2.59) hold for every u ∈ C ∞ ( B R ). Proof.
For u ∈ C ∞ ( B R ), d = dist ( supp u, ∂B R ) > , d = R (cid:12)(cid:12)(cid:12) p − δp − (cid:12)(cid:12)(cid:12) − ( R − d ) (cid:12)(cid:12)(cid:12) p − δp − (cid:12)(cid:12)(cid:12) > , we obtain these statements from the following estimates35 for 0 < δ < p , δ < n Z B R | u | p | x | ( δ − pp − (cid:12)(cid:12)(cid:12) R p − δp − − | x | p − δp − (cid:12)(cid:12)(cid:12) p dx ≤ (cid:18) sup | u | d (cid:19) p Z S Z R − d ρ n − ρ ( δ − pp − dρdθ = ω n (cid:18) sup | u | d (cid:19) p (cid:18) n − δ + p − δp − (cid:19) − ( R − d ) n − δ + p − δp − < ∞ where ω n = meas S , Z B R | u | p | x | δ (cid:12)(cid:12)(cid:12) R p − δp − − | x | p − δp − (cid:12)(cid:12)(cid:12) p − dx ≤ (sup | u | ) p d p − Z S Z R − d ) ρ n − ρ δ dρdθ = ω n (cid:18) sup | u | d (cid:19) p ( n − δ ) − ( R − d ) n − δ < ∞ ; • for 1 < p < δ < n Z B R | u | p | x | ( δ − pp − (cid:12)(cid:12)(cid:12) R p − δp − − | x | p − δp − (cid:12)(cid:12)(cid:12) p dx = R ( δ − p ) pp − Z B R | u | p | x | p (cid:12)(cid:12)(cid:12) R δ − pp − − | x | δ − pp − (cid:12)(cid:12)(cid:12) p dx ≤ ω n n − p (cid:18) sup | u | d (cid:19) p ( R − d ) n − p R ( δ − p ) pp − < ∞ , Z B R | u | p | x | δ (cid:12)(cid:12)(cid:12) R p − δp − − | x | p − δp − (cid:12)(cid:12)(cid:12) p − dx = R ( δ − p ) pp − Z B R | u | p | x | p (cid:12)(cid:12)(cid:12) R δ − pp − − | x | δ − pp − (cid:12)(cid:12)(cid:12) p − dx ≤ ω n n − p (sup | u | ) p d p − ( R − d ) n − p R ( δ − p ) pp − < ∞ ; • for p = δ < n Z B R | u | p | x | p (cid:12)(cid:12)(cid:12) ln R | x | (cid:12)(cid:12)(cid:12) p dx ≤ (sup | u | ) p ω n (cid:16) ln RR − d (cid:17) p Z R − d ρ n − p − dρ = (sup | u | ) p ω n (cid:16) ln RR − d (cid:17) p R n − p n − p < ∞ , Z B R | u | p | x | p (cid:12)(cid:12)(cid:12) ln R | x | (cid:12)(cid:12)(cid:12) p − dx ≤ (sup | u | ) p ω n (cid:16) ln RR − d (cid:17) p − Z R − d ρ n − p − dρ = (sup | u | ) p ω n (cid:16) ln RR − d (cid:17) p − ( R − d ) n − p n − p < ∞ . roposition 2.5. For δ ∈ (0 , n ) and all functions u ∈ W ,p ( B R ) the following Hardyinequalities hold:(i) for δ = p , p < n Z B R |∇ u | p dx ≥ (cid:12)(cid:12)(cid:12)(cid:12) p − δp (cid:12)(cid:12)(cid:12)(cid:12) p Z B R | u | p | x | ( δ − pp − (cid:12)(cid:12)(cid:12) R p − δp − − | x | p − δp − (cid:12)(cid:12)(cid:12) p dx + ( n − δ ) (cid:12)(cid:12)(cid:12)(cid:12) p − δp (cid:12)(cid:12)(cid:12)(cid:12) p − Z B R | u | p | x | δ (cid:12)(cid:12)(cid:12) R p − δp − − | x | p − δp − (cid:12)(cid:12)(cid:12) p − dx. (2.61) By arguments of continuity inequality (2.61) is also true in the case of δ = n and δ = 0 .(ii) for δ = p Z B R |∇ u | p dx ≥ (cid:18) p − p (cid:19) p Z B R | u | p | x | p (cid:12)(cid:12)(cid:12) ln R | x | (cid:12)(cid:12)(cid:12) p dx + (cid:18) p − p (cid:19) p − | n − p | Z B R | u | p | x | p (cid:12)(cid:12)(cid:12) ln R | x | (cid:12)(cid:12)(cid:12) p − dx. (2.62) Proof.
With the expressions (2.58), (2.59), applying (2.57) we obtain (i) and (ii).It is important to mention that in the new Hardy inequalities (2.61) and (2.62) for δ ∈ (1 , n ) the constants (cid:12)(cid:12)(cid:12)(cid:12) p − δp (cid:12)(cid:12)(cid:12)(cid:12) p and (cid:18) p − p (cid:19) p , correspondingly, at their leading termsin the right-hand side are optimal. This follows from Proposition 2.6 below. Proposition 2.6. If p > , n ≥ , < δ < n then for < | x | < R the followinginequalities hold:(i) for δ = p (cid:12)(cid:12)(cid:12)(cid:12) p − δp (cid:12)(cid:12)(cid:12)(cid:12) p | x | (1 − δ ) pp − (cid:12)(cid:12)(cid:12) R p − δp − − | x | p − δp − (cid:12)(cid:12)(cid:12) − p ≥ (cid:18) p − p (cid:19) p ( R − | x | ) − p ; (2.63) (ii) for δ = p (cid:18) p − p (cid:19) p | x | − p (cid:12)(cid:12)(cid:12)(cid:12) ln R | x | (cid:12)(cid:12)(cid:12)(cid:12) − p ≥ (cid:18) p − p (cid:19) p ( R − | x | ) − p . (2.64) Proof. (i) For p > δ inequality (2.63) is equivalent to the estimate g ( r ) = ( p − δ )( R − r ) − ( p − r δ − p − (cid:16) R p − δp − − r p − δp − (cid:17) ≥ , for 0 ≤ r ≤ R. (2.65)Since g ( R ) = 0 and g ′ ( r ) = − ( δ − r δ − pp − (cid:16) R p − δp − − r p − δp − (cid:17) ≤ g ( r ).37or 1 < p < δ < n estimate (2.63) is equivalent to (cid:18) δ − pp (cid:19) p r (1 − δ ) pp − r ( δ − p ) pp − R ( δ − p ) pp − (cid:16) R δ − pp − − r δ − pp − (cid:17) − p ≥ (cid:18) p − p (cid:19) p ( R − r ) − p , or h ( r ) = ( δ − p ) R δ − pp − ( R − r ) − ( p − r (cid:16) R δ − pp − − r δ − pp − (cid:17) ≥ , for 0 ≤ r ≤ R. Since h ( R ) = 0 and h ′ ( r ) = − ( δ − (cid:16) R δ − pp − − r δ − pp − (cid:17) ≤ , inequality (2.63) follows from the monotonicity of h ( r ).(ii) Inequality (2.64) is equivalent to z ( r ) = R − r − r ln Rr ≥ , for 0 < r < R. (2.66)Since z ( R ) = 0 and z ′ ( r ) = − ln Rr <
0, for r ∈ (0 , R ), the inequality (2.66) is a conse-quence of the monotonicity of z ( r ).However, for Hardy inequality Z B R |∇ u | p dx ≥ (cid:18) p − p (cid:19) p Z B R | u | p ( R − | x | ) p dx, the constant (cid:18) p − p (cid:19) p is optimal, see (1.2) with α = 0 and Ω = B R . The same optimalityis also true for (2.61) and (2.62). In this section we prove general Hardy inequalities with singular at zero and on the bound-ary ∂ Ω weights is proved. The Hardy constant is optimal when Ω is a ball. The section isbased on Fabricant et al. [34].Let Ω ⊂ R n , n ≥ ⊂ Ω ∗ and there exists a positive function λ ∈ C , (Ω ∗ ) , λ ( x ) > , such that | x | < λ ( x ) and h x, ∇ λ i ≤ x ∈ Ω ∗ , (3.1)where h , i denotes the scalar product in R n . If Ω is convex or star-shaped domain withrespect to some interior ball centered at zero, then Ω satisfies (3.1), see Section 1.1.8 inMaz’ja [80], so one can take Ω ∗ = Ω. When Ω is an arbitrary domain, then its star-shapedenvelope with respect to some fixed interior ball (i.e., the intersection of all star-shapeddomains with respect to the fixed ball containing Ω) can be taken as Ω ∗ . Further on, wesuppose that 0 ∈ Ω.For α, β ∈ R , p > βp > αp ≤ n + βp − k = n − αpβp − g ( s ) = − s k k for k = 0ln 1 s for k = 0 (3.2)38nd the constant γ = βp − p . Note that γ > k ≥ −
1. For s ( x ) = | x | λ ( x ) let usconsider the non-negative weights v ( x ) = | x | − α | g ( s ( x )) | − β , w ( x ) = | x | − α | g ( s ( x )) | − β . (3.3)The function v is singular at the origin when k ≥ α > k ∈ [ − , α > k ( β −
1) + 1. The function w ( x ) is singular at 0 when k ≥ α > k ∈ [ − ,
0) for α > − βk . Moreover, v and w are singular on the whole boundary if ∂ Ω ∗ = ∂ Ω = { x : | x | = λ ( x ) } . Otherwise, if ∂ Ω ∗ = ∂ Ω, the functions v and w are singularonly on a part of the boundary.Let the space W ,p ,v (Ω) be the completion of C ∞ (Ω) functions with respect to the norm (cid:18)Z Ω v p |∇ u | p dx (cid:19) /p < ∞ , see Lemma 3.1.The aim of this section is to prove the following new Hardy inequality with doublesingular weights. Theorem 3.1.
For every u ∈ W ,p ,v (Ω) the following inequality holds Z Ω v p |∇ u | p dx ≥ γ p Z Ω w p | u | p dx, (3.4) where γ = βp − p . At the beginning, let us analyze condition (3.1) and simplify it in polar coordinates.If S = { x ∈ R n : | x | = 1 } is the unit sphere and ρ = | x | , θ = x | x | ∈ S , the property of λ ( x ), h x, ∇ λ i ≤ ρλ ρ ≤
0, where λ = λ ( ρ, θ ).Indeed, h x, ∇ λ i = ρλ ρ because ∂λ∂x j = x j ρ λ ρ + ∂λ∂θ k (cid:18) δ jk ρ − x j x k ρ (cid:19) , and n X l,j =1 ∂λ∂θ l x j (cid:18) ∂δ jl ρ − x j x l ρ (cid:19) = 0 . The proof of the Theorem is based on Theorem 2.1 and Lemma 3.1 which clarifies theproperties of the weights w and v and the definition of the weighted Sobolev space W ,p ,v (Ω). Lemma 3.1.
Functions w ( x ) and v ( x ) belong to L ploc (Ω) .Proof. Since the functions v and w are singular at 0 and on ∂ Ω, it is enough to proveLemma 3.1 only in a small ball containing 0. For this purpose let us fix ε ∈ (0 ,
1) and wewill prove that w ( x ) is an L p function in a small ball B σ with radius σ ∈ (0 ,
1) centeredat zero, B σ ⊂ Ω, so that ρλ < − ε (3.5)for ρ < σ . 39n polar coordinates, for k = 0 we have Z B σ w p dx = Z S Z σ H ( ρ ) dρdθ, where H ( ρ ) = ρ n − − αp − ( ρλ ) k k ! − βp . We define the constant C k,ε = (cid:18) − (1 − ε ) k k (cid:19) − βp ω n , where ω n is the measure of the unitsphere S in R n , and, let us consider all possibilities for the sign of k :(a) k >
0, i.e., n > αp . Then from ∂∂ρ − (cid:0) ρλ (cid:1) k k ! − βp > Z S Z σ H ( ρ ) dρdθ ≤ Z S Z σ C k,ε ω n ρ n − − αp dρdθ = C k,ε σ n − αp n − αp < ∞ . (b) k <
0, i.e., n < αp . In this case H ( ρ ) = (cid:16) ρλ (cid:17) − kβp ρ n − − αp − ( ρλ ) | k | | k | ! − βp = λ kβp ρ αp − nβp − − − ( ρλ ) | k | | k | ! − βp , and then Z S Z σ H ( ρ ) dρdω ≤ C | k | ,ε βp − αp − n σ αp − nβp − sup θ ∈ S λ kβp ( σ, θ ) < ∞ , because λ is a decreasing function on ρ and kβp < k = 0, i.e., n = αp . Then the simple computations taking into account themonotonicity of λ ( ρ, θ ) and (3.5) give the following chain of inequalities λ ( ρ, θ ) ρ ≥ λ ( σ, θ ) ρ > − ε > , for every ρ < σ , and 0 < (cid:18) ln λ ( ρ, θ ) ρ (cid:19) − βp ≤ (cid:18) ln λ ( σ, θ ) ρ (cid:19) − βp . Hence from (3.2) and the choice of H ( ρ ) we get Z S Z σ H ( ρ ) dρdθ = Z S Z σ (cid:18) ln λρ (cid:19) − βp dρρ dθ ≤ Z S Z σ (cid:18) ln λ ( σ, θ ) ρ (cid:19) − βp dρρ dθ = 1 βp − Z S (cid:18) ln λ ( σ, θ ) σ (cid:19) − βp dθ < ∞ , βp > v ( x ) ∈ L ploc (Ω) is analogous for k ≥ v ( x ) at theorigin is weaker than the singularity of w ( x ).As for k <
0, i.e., n < αp , we obtain Z B σ v p dx = Z S Z σ H ( ρ ) dρdθ, and H ( ρ ) = ρ n − − α ) p − ( ρλ ) k k ! (1 − β ) p = ρ n − − α ) p + kp (1 − β ) λ kp ( β − − ( ρλ ) | k | | k | ! (1 − β ) p . Hence Z B σ v p dx ≤ σ n +(1 − α ) p + k (1 − β ) p n + (1 − α ) p + k (1 − β ) p − ( ρλ ) | k | | k | ! (1 − β ) p λ k ( β − p ω n < ∞ , because λ < ∞ and − ( ρλ ) | k | | k | ! (1 − β ) p ≤ − ( σλ ) | k | | k | ! (1 − β ) p for β > , − ( ρλ ) | k | | k | ! (1 − β ) p ≤ β ∈ (cid:20) p , (cid:21) ,n + (1 − α ) p + k (1 − β ) p > pβp − βp − n − αp ) ≥ , due to the condition np < α ≤ n + βp − p , as well as (1 − β ) p < p − Proof of Theorem 3.1.
Without loss of generality we can suppose that u ∈ C ∞ (Ω) ∩ W ,p ,v (Ω).First, let us consider the case α = np , i.e., k = 0. By (3.1) it holds that h x, ∇ g ( s ( x )) i = − s ( x ) k − (cid:18) | x | λ ( x ) − | x | λ ( x ) h x, ∇ λ ( x ) i λ ( x ) (cid:19) ≤ − s ( x ) k = kg ( s ( x )) − . (3.6)For the vector function ff ( x ) = x | x | α ( p − | g ( s ( x )) | − β ( p − − g ( s ( x )) ,
41e have f ( x ) v ( x ) = x | x | αp | g ( s ( x )) | − βp g ( s ( x )) and | f ( x ) | p ′ = | x | − αp | g ( s ( x )) | − βp ≡ w ( x ) p . (3.7)Since βp > f ( x ) v ( x ))= n − αp | x | αp | g ( s ( x )) | − βp g ( s ( x )) + 1 − βp | x | αp | g ( s ( x )) | − βp h x, ∇ g ( s ( x )) i≥ | x | αp | g ( s ( x )) | βp [( n − αp ) g ( s ( x )) + (1 − βp )( kg ( s ( x )) − βp − w p . (3.8)The last equality in (3.8) holds because ( n − αp ) + k (1 − βp ) = 0.Thus for the functions f ( x ) = − f ( x ) v ( x ) , v ( x ) = v − p ′ ( x ) , w ( x ) = p ( β − w p ( x ) , from (3.7) and (3.8) we obtain − div f − ( p − v | f | p ′ = div( f v ) − ( p − | v | − p ′ | f | p ′ | v | p ′ ≥ ( βp − w p − ( p − w p = p ( β − w p = p ( β − w . (3.9)Moreover, the following identities hold in Ω ε = Ω \ B ε , B ε = {| x | ≤ ε } , supp u ⊂ Ω ⊂ Ω, B ε ⊂ Ω L ( u ) = Z Ω ε v − p (cid:12)(cid:12)(cid:12)(cid:12) h f , ∇ u i| f | (cid:12)(cid:12)(cid:12)(cid:12) p dx ≤ Z Ω ε v p |∇ u | p dx ; K ( u ) = Z Ω ε v | f | p ′ | u | p dx = Z Ω ε | f | p ′ | u | p dx = Z Ω ε w p | u | p dx ; N ( u ) = Z Ω ε w | u | p dx = p ( β − Z Ω ε w p | u | p dx ; K ( u ) = Z Γ + h f , η i| u | p dS, (3.10)where η is outward normal vector to ∂ Ω ε . Note that Z ∂ Ω h f , η i| u | p dS = 0 , and Z ∂B ε h f , η i| u | p dS ≥ , because h f , η i = | x | − α ( p − | g ( s ( x )) | − β ( p − − g ( s ( x )) η ( x ) > ∂B ε .42rom (3.9), (3.10) and Corollary 2.1, we obtain Z Ω ε v p |∇ u | p dx ≥ L ( u ) ≥ (cid:18) p (cid:19) p | K ( u ) + ( p − K ( u ) + N ( u ) | p K p − ( u ) ≥ (cid:18) p (cid:19) p | ( p − K ( u ) + N ( u ) | p K p − ( u ) = (cid:18) p (cid:19) p (cid:12)(cid:12)(cid:12) ( p − p ( β − R Ω ε w p | u | p dx (cid:12)(cid:12)(cid:12) p (cid:16)R Ω ε w p | u | p dx (cid:17) p − ≥ (cid:18) pβ − p (cid:19) p Z Ω ε w p | u | p dx. After the limit ε → α = np , i.e., k = 0, it is enough to replace inequality (3.6) withinequality h x, ∇ g ( s ( x )) i = − λ ( x ) | x | (cid:28) x, ∇ (cid:18) | x | λ ( x ) (cid:19)(cid:29) = λ ( x ) | x | (cid:18) | x | λ ( x ) h x, ∇ λ ( x ) i λ ( x ) − | x | λ ( x ) (cid:19) ≤ − , from (3.1). The rest of the proof is similar to the case α = np . γ p The optimality of γ p for Ω ∗ = Ω = { x : | x | < λ } , λ = const , is guaranteed by the followingtheorem. However, the question whether the inequality (3.4) is sharp in W ,p ,v (Ω) is stillopen. Theorem 3.2.
For every ε, < ε < there exists u ε ∈ W ,p ,v (Ω) such that for L ( u ε ) = Z Ω v p |∇ u ε | p dx and R ( u ε ) = Z Ω w p | u ε | p dx , we have γ p ≤ L ( u ε ) R ( u ε ) ≤ (1 + ε ) p γ p . (3.11)In the proof of the optimality of the constant γ p in inequality (3.4) we will choose thefunction u ε so that the ratio of the left-hand side to the right-hand side of (3.4) is in theinterval [ γ p , (1 + ε ) p γ p ]. In addition, one have to show that both sides of (3.4) are finite. Proof of Theorem 3.2.
For simplicity we suppose that λ ≡ B . In polarcoordinates ( ρ, θ ) with the functions v and w defined in (3.3) and for functions u dependingonly on ρ , the inequality (3.4) becomes L ( u ) = ω n Z ρ n − − p ( α − (cid:12)(cid:12)(cid:12)(cid:12) − ρ k k (cid:12)(cid:12)(cid:12)(cid:12) p − βp | u ρ | p dρ ≥ γ p ω n Z ρ n − − pα (cid:12)(cid:12)(cid:12)(cid:12) − ρ k k (cid:12)(cid:12)(cid:12)(cid:12) − βp | u | p dρ = γ p R ( u ) , (3.12)43here ω is the measure of the unite sphere S in R n .Let us fix 0 < ε <
1. In order to prove (3.11), we choose u ε for different cases of k asfollows:(a) Let k >
0, i.e., n > αp . For u ε = (1 − ρ k ) γ (1+ ε ) ρ − γk (1 − ε ) , we have( u ε ) ρ = − γk (1 + ε )(1 − ρ k ) γ (1+ ε ) − ρ − γk (1 − ε )+ k − − γk (1 − ε )(1 − ρ k ) γ (1+ ε ) ρ − γk (1 − ε ) − = − γk (1 − ρ k ) γ (1+ ε ) − ρ − γk (1 − ε ) − [(1 + ε ) ρ k + (1 − ε )(1 − ρ k )] . Now comparing the functions in the left-hand and right-hand sides in (3.12) with u = u ε we obtain v ( ρ ) | ( u ε ) ρ ( ρ ) | w ( ρ ) u ε ( ρ ) = kγ (cid:20) − ρ k k ρ (cid:21) (1 − ρ k ) − ρ − (2 ερ k + 1 − ε ) ≤ (1 + ε ) γk (cid:20) − ρ k k ρ (cid:21) (cid:20) ρ (1 − ρ k ) (cid:21) ≤ (1 + ε ) γ, and hence L ( u ε ) R ( u ε ) ≤ (1 + ε ) p γ p . (3.13)Note that R ( u ε ) = ω n k βp Z ρ n − αp − − γkp (1 − ε ) (1 − ρ k ) γp (1+ ε ) − βp dρ < ∞ , since n − αp − − γkp (1 − ε ) = ε ( n − αp ) − > − γp (1 + ε ) − βp = ε ( βp − − > − k > k <
0, i.e., n < αp . In this case we define u ε = (1 − ρ − k ) γ (1+ ε ) . Similarcalculations as in (a) give us( u ε ) ρ = kγ (1 + ε ) ρ − k − (1 − ρ − k ) γ (1+ ε ) − , and v ( ρ ) | ( u ε ) ρ | w ( ρ ) u ε ( ρ ) = (1 + ε ) γk (cid:20) | − ρ k | k ρ (cid:21) [ ρ − k − (1 − ρ − k ) − ] = (1 + ε ) γ. Hence L ( u ε ) R ( u ε ) = (1 + ε ) p γ p , and with (3.4) we obtain inequality (3.11).44t remains to check that R ( u ε ) < ∞ . Indeed, R ( u ε ) = ω n Z ρ n − αp − (1 − ρ − k ) γp (1+ ε ) (cid:18) ρ k − | k | (cid:19) − βp dρ = ω n | k | βp Z ρ n − αp − βpk − (1 − ρ | k | ) γp (1+ ε ) − βp dρ < ∞ , because n − αp − βpk − αp − nβp − − > − , and γp (1 + ε ) − βp = ε ( βp − − > − . (c) Let k = 0, i.e., n = αp . We define for a fixed µ the function 0 < µ < u ε = (cid:18) ln 1 ρ (cid:19) γ (1+ ε ) for µ < ρ < , (cid:18) ln 1 µ (cid:19) γ (1+ ε ) for 0 ≤ ρ ≤ µ. Then ( u ε ) ρ = − γ (1 + ε ) 1 ρ (cid:18) ln 1 ρ (cid:19) γ (1+ ε ) − , for µ < ρ < , , for 0 ≤ ρ ≤ µ, and for v ( ρ ) = ρ − α (cid:18) ln 1 ρ (cid:19) − β , w ( ρ ) = ρ − α (cid:18) ln 1 ρ (cid:19) − β , it follows that v ( ρ ) | ( u ε ) ρ | w ( ρ ) u ε = γ (1 + ε ) (cid:20) ρ ln 1 ρ (cid:21) " ρ (cid:18) ln 1 ρ (cid:19) − = γ (1 + ε ) for µ < ρ < , , for 0 ≤ ρ ≤ µ. Hence we get L ( u ε ) R ( u ε ) ≤ (1 + ε ) p γ p , and combining with (3.4) we obtain inequality (3.11).45et us check that R ( u ε ) is finite. Simple computations give us R ( u ε ) = ω n (cid:18) ln 1 µ (cid:19) pγ (1+ ε ) Z µ (cid:18) ln 1 ρ (cid:19) − βp dρρ + ω n Z µ (cid:18) ln 1 ρ (cid:19) pγ (1+ ε ) − βp dρρ = ω n (cid:18) ln 1 µ (cid:19) pγ (1+ ε ) (cid:16) ln ρ (cid:17) − βp | µ βp − − ω n (cid:16) ln ρ (cid:17) pγ (1+ ε ) − βp +1 (cid:12)(cid:12) µ pγ (1 + ε ) − βp + 1= ω n (cid:18) ln 1 µ (cid:19) pγ (1+ ε ) − βp +1 (cid:20) βp − pγ (1 + ε ) − βp + 1 (cid:21) = ω n (cid:18) βp − (cid:19) (cid:18) εε (cid:19) (cid:18) ln 1 µ (cid:19) pγ (1+ ε ) − βp +1 < ∞ , because pγ (1 + ε ) − βp + 1 = ε ( βp − > − βp < . The examples below illustrate that Theorem 3.1 provides a new correction term in Hardyinequalities for weights with one type of singularity either at 0 or on ∂ Ω. Let us recallthat the classical Hardy inequality for p = 2, n = 3 does not attain equality on func-tions from H ( B ), B is the unit ball, it allows the so-called correction term A ( u ) = (cid:18) (cid:19) Z B Q ( x ) u dx , see (1.15), i.e., Z B |∇ u | dx − Z B | u | | x | dx ≥ Z B Q ( x ) | u | dx, for u ∈ H ( B ) . (3.14)In Adimurthi et al. [3], Alvino et al. [6], Brezis and Vazquez [26], Filippas and Tertikas[42], inequality (3.14) is proved for different radial symmetric weights Q ( x ) = q ( r ), r = | x | : • q ( r ) = const in Brezis and Vazquez [26]; • q ( r ) = 1 r l X j =1 j Y i =1 ln ( j ) ρr , ρ = (max | x | ) e e e..e − l − times , where ln ( m ) ( . ) = ln(ln ( m − ( . ))in Adimurthi et al. [3]; • q ( r ) = 1 r ∞ X i =1 X ( rD ) · · · X i ( rD ), X ( t ) = (1 − ln( t )) − , X m = X ( X m − ( t )), D ≥ • q ( r ) = 1( r ln r ) in Alvino et al. [6]. 46eneral characteristics of the possible Q ( x ) were given in Ghoussoub and Moradifam [50],Theorem 1. It was shown that (3.14) is valid for Q ( x ) with decreasing q ( r ) if and only ifthe ordinary differential equation y ′′ ( r ) + y ′ ( r ) r + q ( r ) y ( r ) = 0 , (3.15)has a positive solution on (0 , Q ( x ). Example 3.1.
Let Ω = B , λ ( x ) = 1, α = β = 1, n = 3, p = 2. Then Ω ∗ = B , k = 1, γ = 12 v = 1, w = | x | − (1 − | x | ) − and (3.4) takes the form of Z B |∇ u | dx ≥ Z B | u | | x | (1 − | x | ) dx, for u ∈ H ( B ) . (3.16)The weight of the right-hand side of (3.16) has singularities at zero and on ∂B incontrast to the weight in the papers of Adimurthi et al. [3], Brezis and Vazquez [26],Filippas and Tertikas [42].Factorize 1 | x | (1 − | x | ) = 1 | x | + 2 − | x || x | (1 − | x | ) then inequality (3.16) takes the form of (3.14) with a kernel Q ( x ) = 2 − | x || x | (1 − | x | ) , i.e., Z B |∇ u | dx − Z B | u | | x | dx ≥ Z B (2 − | x | ) | u | | x | (1 − | x | ) dx, for u ∈ H ( B ) . (3.17)Here Q ( x ) = q ( r ), where q ( r ) is radially symmetric for r = | x | ∈ (0 , q ( r ) → ∞ for r → r →
1. The inequality (3.17) is not included in Ghoussoub andMoradifam [50] since the function q ( r ) in (3.15) is not decreasing on (0 , q ( r ) = 2 − rr (1 − r ) is a linear combination ofHypergeometric functions and has no positive solution in the whole interval (0 , B and n = 3 for φ ( r ) = 1, h ( r ) = (cid:18) r − r (cid:19) / ,inequality (1.8) follows from Example 3.1. However, rφ ( r )( h ) ′ ( r ) = r (1 − r ) = const, and (1.9) fails. Moreover, a simple computation shows that for φ ( r ) ≡
1, the weight (cid:12)(cid:12)(cid:12)(cid:12) h ′ ( r ) h ( r ) (cid:12)(cid:12)(cid:12)(cid:12) of the right-hand side of (1.8) cannot be singular both at 0 and at 1 if condition(1.9) is satisfied. 47n Brezis and Marcus [25] the following Hardy inequality Z B |∇ u | dx − Z B | u | (1 − | x | ) dx ≥ Z B Q ( x ) | u | dx, for u ∈ H ( B ) , (3.18)was proved for Q ( x ) = const , see (1.11). If we factorize1 | x | (1 − | x | ) = 1(1 − | x | ) + 1 + | x || x | (1 − | x | ) , inequality (3.16) transforms into (3.18) with Q ( x ) = (1 + | x | ) | x | (1 − | x | ) .In all inequalities obtained in Adimurthi et al. [3], Brezis and Vazquez [26], Brezis andMarcus [25], Filippas and Tertikas [42], the constants are optimal. From Theorem 3.2, itfollows that for Hardy inequality (3.4) the constant γ p is optimal as well.Finally, in Filippas and Tertikas [42], the authors show the validity of (3.4) under therestriction 2 = p < n . In the present section there are no restrictions on p except p > ∗ in (3.1). It is well known that thereare no conditions on Ω for Hardy inequality with singularity at 0 ∈ Ω. However, whenthe singularity is on ∂ Ω then the restrictions about the convexity of the domain or itsgeneralization are always considered. We will give three simple examples for λ ( x ) and Ω ∗ when condition (3.1) holds. Example 3.2.
Let Ω = B λ = { x, | x | < λ } , λ = const > ∗ = Ω. In this casethe weights v ( x ), w ( x ) are singular at 0 and on the whole boundary ∂ Ω = ∂B λ . If forsimplicity α = β = 1, γ = p − p and k = n − pp − = 0, i.e., 1 < p = n , then Hardy inequality(3.4) becomes Z B λ |∇ u | p dx ≥ (cid:18) | n − p | λ k p (cid:19) p Z B λ | u | p | x | p ( λ k − | x | k ) p dx. Example 3.3.
Let Ω be a star-shaped domain with respect to an interior ball centeredat zero, so that Ω ∗ = Ω. In this case we can choose λ = λ ( θ ) where θ is the angularvariable of x and ∂ Ω = { x, | x | = λ ( θ ) } . According to Lemma in section 1.1.8 of Maz’ja[80], λ ( θ ) ∈ C , (Ω) and condition (3.1) and respectively Hardy inequality (3.4) holds.Note that in this case the weights v ( x ) and w ( x ) are singular at zero and on the wholeboundary ∂ Ω. Example 3.4.
Let n > p , α = 1, β = 1, γ = p − p so that k = n − pp − > D ⊂ R n we define H ( D ) = Z D |∇ u | p dx, H ( D ) = C Z D | u | p | x | p dx,H ( D ) = C Z D | u | p ( λ ( x ) − | x | ) p dx. We will define two domains Ω , Ω such that ¯Ω ⊂ Ω, Ω = Ω \ ¯Ω , 0 ∈ Ω and we willshow that H (Ω) ≥ H (Ω ) + H (Ω ), i.e., Z Ω |∇ u | p dx ≥ C Z Ω | u | p | x | p dx + C Z Ω | u | p ( λ ( x ) − | x | ) p dx, (3.19)48ith C = (cid:18) n − pp (cid:19) p , C = (cid:18) p ′ (cid:19) p for u ∈ W ,p (Ω).In order to prove (3.19) let us mention that (cid:18) | x | λ ( x ) (cid:19) k < k > h ( s ) = 1 − s k k − s − + 1, where s = | x | λ ( x ) . Since h ′ ( s ) > ,
1] and its maximum is attained for s = 1. Hence h ( s ) ≤ | x | k − (cid:18) | x | λ ( x ) (cid:19) k ! ≤ λ ( x ) − | x | . (3.20)From k > | x | k − (cid:18) | x | λ ( x ) (cid:19) k ! ≤ | x | k , holds and combining with (3.20) we get | x | k − (cid:18) | x | λ ( x ) (cid:19) k ! ≤ min (cid:18) | x | k , λ ( x ) − | x | (cid:19) , x ∈ Ω . (3.21)For v ( x ) = 1 , w ( x ) = | x | − − (cid:18) | x | λ ( x ) (cid:19) k ! − , applying (3.21) in Ω = { x ∈ Ω : min (cid:18) | x | k , λ ( x ) − | x | (cid:19) = | x | k } , 0 ∈ Ω and Ω = Ω \ ¯Ω correspondingly, we get from (3.4) the chain of inequalities H (Ω) = Z Ω |∇ u | p dx ≥ (cid:18) p − p (cid:19) p Z Ω | u | p | x | p (cid:18) − (cid:16) | x | λ ( x ) (cid:17) k (cid:19) p dx ≥ (cid:18) p − p (cid:19) p ( k ) p Z Ω | u | p | x | p dx + (cid:18) p ′ (cid:19) p Z Ω | u | p ( λ ( x ) − | x | ) p dx = H (Ω ) + H (Ω ) . Note that in (3.19) the constants C and C are optimal for the corresponding classicalcases with single singular weights. However, (3.19) cannot be obtained by summing theclassical Hardy inequalities H (Ω) ≥ H (Ω ) and H (Ω) ≥ H (Ω ) because they are validonly for u ∈ W ,p (Ω ) ∩ W ,p (Ω ).Let us recall that 0 ∈ Ω in Example 3.4 and this is essential for the optimality ofthe constant C in (3.19). Remark that in the case when 0 ∈ ∂ Ω there exists a constant C ′ > C , see Nazarov [82], Pinchover and Tintarev [87].49 Sharp Hardy inequalities with weights singular at an in-terior point
In this section we prove Hardy inequality with weight singular at 0 ∈ Ω ⊂ R n , n ≥ ∂ Ω. Hardy’s constant is optimaland the inequality is sharp due to the additional boundary term. The section is based onFabricant et al. [35].In order to formulate our main results we recall the definition of the trace operator,see Adams [1], Evans [32], Ch. 5.5 and Maz’ja [80], Ch. 1.4.5.
Definition 4.1.
For a bounded C smooth domain Ω ⊂ R n , n ≥ and p > thetrace operator T : W ,p (Ω) → L p ( ∂ Ω) is a bounded linear operator, T u = u | ∂ Ω for u ∈ W ,p (Ω) ∩ C ( ¯Ω) and k T u k L p ( ∂ Ω) ≤ C ( p, Ω) k u k W ,p (Ω) for u ∈ W ,p (Ω) . Let us consider the following inequality Z Ω | x | l (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u ( x ) i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx ≥ (cid:12)(cid:12)(cid:12)(cid:12) p − l − np (cid:12)(cid:12)(cid:12)(cid:12) p Z Ω | x | l − p | u ( x ) | p dx, u ∈ W ,pl, (Ω) , (4.1)for the constant l = p − n , where p > n ≥ C is smooth, bounded domain Ω ⊂ R n ,0 ∈ Ω. Here W ,pl, (Ω) is the completion of C ∞ (Ω) functions with respect to the norm (cid:18)Z Ω | x | l |∇ u ( x ) | p dx (cid:19) /p < ∞ , (4.2)satisfying the conditionlim ε → ε l − p +1 Z S ε | T u | p dS = 0 , S ε = { x ∈ Ω; | x | = ε } . (4.3)The constant (cid:12)(cid:12)(cid:12)(cid:12) p − l − np (cid:12)(cid:12)(cid:12)(cid:12) p in (4.1) is optimal but inequality (4.1) is not sharp in W ,pl, (Ω),see Hardy et al. [58] for n = 1. That is why we introduce a more general class of functionsˆ W ,pl (Ω), without any restrictions of u on ∂ Ω, and define an additional term dependingon the trace of u on ∂ Ω.We denote ∂ Ω − = { x ∈ ∂ Ω : sgn( p − l − n ) h x, η i < } , where η is the unit outwardnormal vector to ∂ Ω, and consider the norm (cid:18)Z Ω | x | l |∇ u ( x ) | p dx (cid:19) /p + (cid:18) p − p (cid:19) p − Z ∂ Ω − |h x, η i|| x | l − p | u ( x ) | p dS < ∞ , (4.4)see Maz’ja [80], Ch. 1.1.15 in the case | ∂ Ω − | 6 = ∅ and Ch. 1.1.6 in the case | ∂ Ω − | = ∅ .Let us mention that | ∂ Ω − | = 0 if and only if p > l + n and Ω is a star-shaped domainwith respect to the origin, according to Definition 4.2 belowWe define ˆ W ,pl (Ω) as the completion of C ∞ (Ω) ∩ C ( ¯Ω) functions in the norm (4.4)which satisfy (4.3). Note that for p − l − n < u ∈ C ∞ (Ω) ∩ C ( ¯Ω), while for p − l − n >
0, condition (4.3) requires u (0) = 0. Actually,ˆ W ,pl (Ω) for p − l − n > C ∞ (Ω) ∩ C ( ¯Ω) functions in the norm (4.4)which are equal to zero near the origin (see Remark 4.1 below).50 heorem 4.1. Suppose Ω ⊂ R n is a bounded domain with C smooth boundary and ∈ Ω . Then for every constant l = p − n , p > , n ≥ and for every u ∈ ˆ W ,pl (Ω) thefollowing inequality holds (cid:18)Z Ω | x | l (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u ( x ) i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx (cid:19) /p ≥ (cid:12)(cid:12)(cid:12)(cid:12) p − l − np (cid:12)(cid:12)(cid:12)(cid:12) (cid:18)Z Ω | x | l − p | u ( x ) | p dx (cid:19) /p + 1 p sgn ( p − l − n ) Z ∂ Ω | x | l − p h x, η i| T u | p dS (cid:18)Z Ω | x | l − p | u ( x ) | p dx (cid:19) − /p ′ , (4.5) where p + 1 p ′ = 1 and η is the unit outward normal vector to ∂ Ω . Remark 4.1.
The standard definition of the space of functions in (4.5) for p − l − n > C ∞ (Ω) ∩ C ( ¯Ω) functions, with respect to the norm (4.4), which arezero near the origin. However, applying Hardy inequality (4.5) to the ball B ε = { x ∈ Ω : | x | < ε } , after the limit ε →
0, we get from Z B ε | x | l |∇ u | p dx → Z B ε | x | l − p | u | p dx → . (4.6)Hence from (4.5) and (4.6) it follows that Z S ε | x | l − p h x, η i| u | p dS = ε l − p +1 Z S ε | u | p dS → , i.e., (4.3) is satisfied. In this way we get the same space ˆ W ,pl (Ω). Proof of Theorem 4.1.
In order to prove Theorem 4.1, let us introduce the notations f ( x ) = sgn( p − l − n ) | x | l − p x, v ( x ) = | p − l − n | p − | x | l/ (1 − p ) (4.7)and L ( u ) = Z Ω v − p (cid:12)(cid:12)(cid:12)(cid:12) h f, ∇ u i| f | (cid:12)(cid:12)(cid:12)(cid:12) p dx = (cid:18) | p − l − n | p − (cid:19) − p Z Ω | x | l (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx ; K ( u ) = Z Ω v | f | p ′ | u | p dx = | p − l − n | p − Z Ω | x | l − p | u | p dx ; K ( u ) = Z ∂ Ω h f, η i| T u | p dS = sgn( p − l − n ) Z ∂ Ω | x | l − p h x, η i| T u | p dS. (4.8)Simple computations show that f and v satisfy the equality − div f − ( p − v | f | p ′ = 0 , in Ω \{ } , (4.9)because − ( p − v | f | p ′ = −| p − l − n || x | l − p and − div f = − sgn( p − l − n ) h n | x | l − p + ( l − p ) | x | l − p i = | p − l − n || x | l − p . u ∈ C ∞ (Ω) ∩ C ( ¯Ω) satisfying(4.3), (4.4).For every small positive constant ε we apply Corollary 2.1 in Ω ε = Ω \ B ε , B ε = {| x | ≤ ε } , for w ( x ) ≡ ε → N ( u ) = 0, in notations (4.8) we obtain L ( u ) ≥ (cid:18) p (cid:19) p | K ( u ) + ( p − K ( u ) | p K p − ( u ) . Now using the choice of f in (4.7) we obtain (4.5) for u ∈ C ∞ (Ω) ∩ C ( ¯Ω).By standard approximation argument (see Maz’ja [80], Ch. 1.1.15 and Ch. 1.1.6), weget (4.5) for every u ∈ ˆ W ,pl (Ω). Remark 4.2.
By means of the simple inequality | z | p ≥ pz for every z and p > L ( u ) ≥ (cid:18) p − p (cid:19) p K ( u ) (cid:12)(cid:12)(cid:12)(cid:12) p − K ( u ) K − (cid:12)(cid:12)(cid:12)(cid:12) p ≥ (cid:18) p − p (cid:19) p K ( u ) + (cid:18) p − p (cid:19) p − ( K , + ( u ) − K , − ( u )) , where K , + = Z ∂ Ω \ ∂ Ω − | x | l − p |h x, η i|| T u | p dS, K , − = Z ∂ Ω − | x | l − p |h x, η i|| T u | p dS. So, for u ∈ ˆ W ,pl (Ω) we obtain the ‘linear’ form of Hardy inequality (4.5) L ( u ) + (cid:18) p − p (cid:19) p − K , − ( u ) ≥ (cid:18) p − p (cid:19) p K ( u ) + (cid:18) p − p (cid:19) p − K , + ( u ) . Theorem 4.2.
Under the assumptions of Theorem 4.1 inequality (4.5) is an equality for u k = | x | k Φ (cid:18) x | x | (cid:19) (4.10) for every smooth function Φ and every constant k > p − l − np , such that u k ∈ ˆ W ,pl (Ω) ,i.e., (4.5) is sharp in ˆ W ,pl (Ω) .Proof. From Theorem 2.2, it follows that (4.9) is an equality if (2.14)–(2.16) hold, i.e., if(4.9) and u h f, ∇ u i = | u h f, ∇ u i| , h f, ∇ u i = k p v | f | p ′ u, (4.11)are fulfilled for a.e. x ∈ Ω and for some constant k ≥
0. From the choice of f and v in(4.7), equation (4.11) is equivalent to sgn ( p − l − n ) u h x, ∇ u i = | u h x, ∇ u i| , h x, ∇ u i = ku, (4.12)52here k = k p − l − np − u = | x | k z ( x ) ∈ ˆ W ,pl (Ω). Simple computations give us that z ( x ) is a solution of thehomogeneous, first order partial differential equation n X i =1 x i z x i = 0 in Ω \{ } . (4.13)The system of characteristic equations of (4.13) becomes˙ x ( t ) = x , . . . , ˙ x n ( t ) = x n , (4.14)and the functions ϕ ( x ) = x | x | , . . . , ϕ n ( x ) = x n | x | are constants along the trajectories of(4.14), i.e., ϕ , . . . , ϕ n are the first integrals of (4.13). Note that only n − z ( x ) = Φ (cid:18) x | x | (cid:19) . Hence function u k ( x ) = | x | k Φ (cid:18) x | x | (cid:19) is a general solution of (4.12) for arbitrary smoothfunction Φ and constant k , such that | x | k Φ ∈ ˆ W ,pl (Ω), i.e., u k satisfies (4.2) and (4.3).Let us check up when condition (4.3) holds. With the change of the variables x = εy we get ε l − p +1 Z S ε | u k | p dS = ε l − p +1 Z S ε | x | kp (cid:12)(cid:12)(cid:12)(cid:12) Φ (cid:18) x | x | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p dS = ε l − p +1+ kp + n − Z | y | =1 | Φ( y ) | p dS. Therefore, (4.3) is satisfied if and only if k > p − l − np and Z | y | =1 | Φ( y ) | p dS < ∞ .We will prove that both sides of (4.5) are finite for all functions u k defined in (4.10).For this purpose it is enough to check that K ( u k ) < ∞ . In fact, for some fixed smallconstant, a ∈ (0 ,
1) and for the ball B a = {| x | < a } , we get the chain of inequalities Z Ω | x | l − p + kp (cid:12)(cid:12)(cid:12)(cid:12) Φ (cid:18) x | x | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p dx = Z B a | x | l − p + kp | Φ | p dx + Z Ω \ B a | x | l − p + kp | Φ | p dx ≤ C Z B a | x | λ dx + C < ∞ , for λ = l − p + kp > − n and some constants 0 < C , C < ∞ . The above inequality followsfrom k > p − l − np .In the following remarks we will compare our result in Theorem 4.1, i.e., inequality(4.5) with the corresponding results about Hardy inequalities with additional boundaryterm in Wang and Zhu [98], Barbatis et al. [15], Berchio et al. [20].53 emark 4.3. Consider the special case of the constants : p = 2, l = − a , a > n − a − > B ⊂ R n . Since (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u i| x | (cid:12)(cid:12)(cid:12)(cid:12) p < |∇ u | p , from (4.5),rising both sides of this inequality to second power and since η = x on ∂B we get Z B | x | − a |∇ u | dx ≥ (cid:18) n − − a (cid:19) (cid:18)Z B | x | − a − | u | dx (cid:19) − n − a − Z ∂B | T u | dS + 14 (cid:18)Z ∂B | T u | dS (cid:19) (cid:18)Z B | x | − a − | u | dx (cid:19) − , u ∈ ˆ W , − a ( B ) . (4.15)In Wang and Zhu [98], the following Hardy inequality with additional boundary term wasproved Z B | x | − a |∇ u ( x ) | dx > (cid:18) n − − a (cid:19) Z B | x | − a − | u ( x ) | dx − n − − a Z ∂B | T u | dS, u ∈ ˆ W , − a ( B ) . (4.16)The constant (cid:18) n − − a (cid:19) in both inequalities (4.15) and (4.16) is optimal. The dif-ference between (4.16) and (4.15) is the additional positive term. Moreover, (4.16) isnot sharp, i.e., it is a strict inequality, while the additional term in (4.15) guarantees itssharpness, i.e., there exists a class of functions ˆ W , − a ( B ) for which (4.15) is an equality. Remark 4.4.
In Barbatis et al. [15] new Hardy inequalities in bounded domains Ω forfunctions H (Ω), see eq. (2.4) in Barbatis et al. [15] are obtained Z Ω |∇ u | dx + c Z ∂ Ω | x | − h x, η i| T u | dS ≥ c ( n − − c ) Z Ω | x | − | u | dx, (4.17)for u ∈ H (Ω), where 0 < c ≤ n −
22 .The inequality (4.5) for the case p = 2 , l = 0 , n > p with a similar transformation asin Remark 4.3 reads Z Ω |∇ u ( x ) | dx ≥ (cid:18) n − (cid:19) Z Ω | x | − | u ( x ) | dx − n − Z ∂ Ω | x | − h x, η i| T u | dS + 14 (cid:18)Z ∂ Ω | x | − h x, η i| T u | dS (cid:19) (cid:18)Z Ω | x | − | u ( x ) | dx (cid:19) − , u ∈ H (Ω) . (4.18)The comparison of (4.18) and (4.17) for the optimal constant c = n −
22 shows that in(4.18) there exists an additional positive term on the right-hand side. Moreover, with this54dditional term inequality (4.18) is sharp, i.e., it is an equality for the class of functions u = u k ∈ H (Ω) defined in (4.10). Remark 4.5.
In Berchio et al. [20], see (13), the following Hardy inequality is studied Z Ω |∇ u | dx + c Z ∂ Ω | T u | dS ≥ h ( c ) Z Ω | x | − | u | dx, u ∈ H (Ω) , (4.19)where c ∈ [0 , C n ], C n ≥ n −
22 , ( C n = n −
22 for Ω = B ) and h ( c ) ∈ " , (cid:18) n − (cid:19) , aredefined as h ( c ) = inf u ∈ H (Ω) \{ } Z Ω |∇ u | dx + c Z ∂ Ω | T u | dS Z Ω | x | − | u | dx . (4.20)By means of positive solutions of the eigenvalue problem under Steklov boundary condi-tions (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ∆ u = h ( c ) | u | | x | , in Ω u η + cu = 0 , on ∂ Ω , see (15) in Berchio et al. [20], it is shown (see Theorem 8 in Berchio et al. [20]) that forthe value of c = n −
22 the infimum in (4.20), i.e., h (cid:18) n − (cid:19) = (cid:18) n − (cid:19) is not achieved. This means that for the optimal constant (cid:18) n − (cid:19) Hardy inequality(4.19) is not sharp.The inequality (4.5) for the case p = 2 , l = 0 , n > p and Ω = B (0) becomes (4.15) inRemark 4.3 for a = 0 and in comparison with (4.19) has an additional positive term in theright-hand side. Moreover, the inequality (4.15) is an equality for the functions defined in(4.10).As a consequence of Theorem 4.1 we get an extension of the classical Hardy inequality(4.1) for functions u in the largest class ˆ W ,pl (Ω), i.e., when u is not necessary zero on thewhole boundary ∂ Ω. We consider the case of domains Ω ⊂ R n , n ≥
2, 0 ∈ Ω which are star-shaped with respectto the origin. Let us recall the definitions of a star-shaped and strictly star-shaped C smooth domains. Definition 4.2.
The domain Ω , ∂ Ω ∈ C is: ) star-shaped domains with respect to the origin if h x, η ( x ) i ≥ , for every x ∈ ∂ Ω , (4.21) where η ( x ) is the unit outward normal vector to ∂ Ω at the point x ∈ ∂ Ω .ii) strictly star-shaped domains with respect to the origin if inequality (4.21) is strict,i.e., h x, η ( x ) i > , for every x ∈ ∂ Ω , (4.22)Let us note that for star-shaped domains the sign of the additional term in (4.5)depends only on the sign of the constant p − l − n . Theorem 4.3.
Suppose Ω is a bounded, star-shaped domain with respect to the origin in R n , n ≥ with C smooth boundary ∂ Ω and ∈ Ω . Then for every p > we have:(i) If p − l − n > , inequality (4.1) is satisfied for every u ( x ) ∈ ˆ W ,pl (Ω) and theconstant p − l − np in (4.1) is optimal.If additionally Ω is a strictly star-shaped domain with respect to the origin, then (4.1)is not sharp in ˆ W ,pl (Ω) .(ii) If p − l − n < , then (4.1) in general does not hold, for example, for functions u k ∈ ˆ W ,pl (Ω) defined in (4.10).Proof. (i) From (4.21) inequality (4.1) holds from (4.5).It is easy to prove that when p − l − n > p − l − np is optimal in (4.1)and (4.5). For this purpose, we will use function u k , defined in (4.10) with k > p − l − np and Φ ≡
1. Since (4.5) is an equality for every u k we get (cid:18) L ( | x | k ) K ( | x | k ) (cid:19) /p = p − l − np + 1 p Z ∂ Ω | x | l − p + kp < x, η > dS × (cid:18)Z Ω | x | l − p + kp dx (cid:19) − → p − l − np for k → p − l − np + 0 . The above limit follows from the inequalities Z Ω | x | l − p + kp dx ≥ Z B a | x | l − p + kp dx = ω n Z a r l − p + kp + n − dr = ω n a l − p + kp + n l − p + kp + n → + ∞ , for k → p − l − np + 0 , and 1 p Z ∂ Ω | x | l − p + kp h x, η i dS → p Z ∂ Ω | x | − n h x, η i dS < ∞ for k → p − l − np + 0 , where ω n is the measure of the unit sphere in R n and B a ⊂ Ω , ∂B a ∩ ∂ Ω = ∅ .56f we suppose that (4.1) is sharp for some function w ( x ) ∈ ˆ W ,pl (Ω) in a strictly star–shaped domain Ω then from (4.1) and (4.5) we have Z ∂ Ω | x | l − p h x, η i| T w | p dS = 0 . Hence due to (4.22) it follows that
T w = 0 for a. e. x ∈ ∂ Ω . This means that (4.1) is also sharp in ˆ W ,pl, (Ω) which proves Theorem 4.3 (i).(ii) If p − l − n <
0, then for u k ( x ) = | x | k we get from Theorem 4.2 and (4.21) Z Ω | x | l (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u k ( x ) i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx = − p Z ∂ Ω | x | l − p h x, η i| u k ( x ) | p dS × (cid:18)Z Ω | x | l − p | u k ( x ) | p dx (cid:19) − /p ′ + | p − l − n | p (cid:18)Z Ω | x | l − p | u k ( x ) | p dx (cid:19) /p < | p − l − n | p (cid:18)Z Ω | x | l − p | u k ( x ) | p dx (cid:19) /p . Hence (4.1) is not satisfied for u = u k ( x ) = | x | k ∈ ˆ W ,pl (Ω) and k > p − l − np whichproves Theorem 4.3 (ii). In order to prove (4.1) without geometry conditions (4.21) or (4.22) as in Sect. 4.1 wespecify the class of functions. Let us introduce the spaces W ,pl, + (Ω), resp. W ,pl, − (Ω), whichare the completion of C ∞ (Ω) ∩ C ( ¯Ω) functions with respect to the norm (4.4), satisfyingin addition (4.3) and (4.23), resp. (4.3) and (4.24): Z ∂ Ω | x | l − p h x, η i| T u | p dS ≥ , (4.23) Z ∂ Ω | x | l − p h x, η i| T u | p dS ≤ . (4.24)Note that obviously the following inclusions hold: W ,pl, (Ω) ⊂ W ,pl, ± (Ω) ⊂ ˆ W ,pl (Ω); W ,pl, + (Ω) ∪ W ,pl, − (Ω) = ˆ W ,pl (Ω) . By means of conditions (4.23) or (4.24) one can control the sign of the additional term ininequality (4.5), and, we have the following result for general domains.
Theorem 4.4.
Suppose Ω is a bounded domain in R n , n ≥ with C smooth boundary, ∈ Ω and p > . Then:(i) Inequality (4.1) holds for every u ∈ W ,pl, (Ω) ;(ii) If p − l − n < , then inequality (4.1) holds for all functions u ∈ W ,pl, − (Ω) ;(iii) If p − l − n > , then inequality (4.1) holds for all functions u ∈ W ,pl, + (Ω) . Theconstant p − l − np is optimal but inequality (4.1) is not a sharp one in W ,pl, + (Ω) . However,an inequality with additional term (4.5) is sharp in W ,pl, + (Ω) .
57n order to prove Theorem 4.4 we need the following Lemma:
Lemma 4.1.
Let Ω be a bounded domain in R n , n ≥ with C smooth boundary ∂ Ω , ∈ Ω and p > . Then identity Z ∂ Ω | x | l − p + kp h x, η i (cid:12)(cid:12)(cid:12)(cid:12) T Φ (cid:18) x | x | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p dS = ( l − p + kp + n ) Z Ω | x | l − p + kp (cid:12)(cid:12)(cid:12)(cid:12) Φ (cid:18) x | x | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p dx > holds for every k > p − l − np and every nontrivial function | x | k Φ (cid:18) x | x | (cid:19) ∈ ˆ W ,pl (Ω) .Proof. If ε > B ε ⊂ Ω, where B ε = { x : | x | < ε } then div (cid:18) | x | l − p + kp x (cid:12)(cid:12)(cid:12)(cid:12) Φ (cid:18) x | x | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p (cid:19) = ( l − p + kp + n ) | x | l − p + kp (cid:12)(cid:12)(cid:12)(cid:12) Φ (cid:18) x | x | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p + | x | l − p + kp h x, ∇| Φ | p i = ( l − p + kp + n ) | x | l − p + kp (cid:12)(cid:12)(cid:12)(cid:12) Φ (cid:18) x | x | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p , for a.e. x ∈ Ω \ B ε . With integration by parts of the above equality in Ω \ B ε , equality (4.25) follows after thelimit ε → Z S ε | x | l − p + kp h x, η i (cid:12)(cid:12)(cid:12)(cid:12) T Φ (cid:18) x | x | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p dS = − ε l − p + kp +1 Z S ε (cid:12)(cid:12)(cid:12)(cid:12) T Φ (cid:18) x | x | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p dS → , as ε → . Proof of Theorem 4.4. (i) The inequality (4.1) is a direct consequence of (4.5). Theoptimality of the constant | p − l − n | p for (4.1) follows in the same way as in Theorem 4.3(i) for (4.5) with p − l − n > p − l − n < z ( x ) ∈ W ,pl, − (Ω), then from (4.1) and (4.5) we get G ( z ) = Z ∂ Ω | x | l − p h x, η i| T z | p dS ≥ . (4.26)Since z ( x ) ∈ W ,pl, − (Ω) from (4.24) it follows that G ( z ) = 0 and (4.1) is sharp in ˆ W ,pl, (Ω)which is impossible.(iii) For p − l − n > u k ( x ) = | x | k , k > p − l − np inequality (4.5) becomes an equality. Since58 k ( x ) ∈ ˆ W ,pl (Ω) it is enough to show that u k ( x ) satisfies (4.23), i.e., u k ( x ) ∈ W ,pl, + (Ω).This follows from Lemma 4.1 for Φ ≡ w ( x ) ∈ W ,pl, + (Ω), then from (4.1) and(4.5) we have G ( w ) ≤ G is defined in (4.26). Since w ( x ) ∈ W ,pl, + (Ω),i.e., G ( w ) ≥
0, then from (4.23) it follows that G ( w ) = 0. This means that (4.5) is sharpfor the function w ( x ) ∈ ˆ W ,pl, (Ω) which is impossible.The optimality of the constant p − l − np follows in the same way as in the proof ofTheorem 4.3 (i). In the present section we prove Hardy inequalities with double singular weights in bounded,star-shaped domains Ω ⊂ R n , n ≥
2. The weights are singular at an interior point andon the boundary of the domain. Hardy’s constant is optimal and the inequality is sharpdue to the additional term, i.e., there exists a non-trivial function for which the inequalitybecomes equality, see Definition 1.1. This section is based on Fabricant et al. [40].In section 4.1, star-shaped domain and a strictly star-shaped domain with respect to0 ∈ Ω are defined, where ∂ Ω ∈ C , see Definition 4.1. Here we use more general Definitions5.1 and 5.2, when ∂ Ω ∈ C . Definition 5.1.
The bounded domain Ω ⊂ R n , n ≥ with C boundary ∂ Ω is star–shapeddomain with respect to a point x ∈ Ω if every ray starting from x intersects the boundary ∂ Ω only at one point. Definition 5.2.
The bounded domain Ω , where ∂ Ω ∈ C is a star-shaped with respect toan interior ball B ε = {| x | < ε } ⊂ Ω if Ω is star-shaped with respect to every point of theball B ε , see Definition 5.1 and Ch. 1.1.6 in Maz’ja [80]. Let Ω = {| x | < ϕ ( x ) } ⊂ R n be a star-shaped domain with respect to a small ball.Here 0 ∈ Ω, n ≥ p >
1, and ϕ is a homogeneous function of the 0-th order. Note that,according to Ch. 1.1.8 in Maz’ja [80], in this case ϕ ( x ) is Lipschitz function on the unitsphere S in R n .We denote by W ,p ( | x | l (1 − p ) , Ω), l ≤ n − p − , l ∈ R , the completion of C ∞ (Ω) functionswith respect to the norm k u k W ,p ( | x | l (1 − p ) , Ω) = (cid:18)Z Ω | x | l (1 − p ) |∇ u | p dx (cid:19) p < ∞ , (5.1)(see Maz’ja [80], Ch.1.1.6).In Theorem 3.1 the following Hardy inequality with double singular weights (3.4) isproved Z Ω | x | l (1 − p ) (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx ≥ (cid:12)(cid:12)(cid:12)(cid:12) m + lp ′ (cid:12)(cid:12)(cid:12)(cid:12) p Z Ω | u | p | x | n − m − l (cid:12)(cid:12)(cid:12)(cid:12) − (cid:16) | x | ϕ (cid:17) m + l (cid:12)(cid:12)(cid:12)(cid:12) p dx, u ∈ W ,p ( | x | l (1 − p ) , Ω) , (5.2)59here we use the notations in Sect. 3: α = 1 + lp ′ , β = 1 , γ = p − p ,v = | x | − lp ′ , w = | m + l || x | − − lp ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:18) | x | ϕ (cid:19) m + l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ,g ( s ( x )) = 1 − s ( x ) m + l m + l , s ( x ) = | x | ϕ ( x ) and m + l = 0 , l ≤ − m. Here m = p − np − = 0, 1 p + 1 p ′ = 1, h ., . i is the scalar product in R n and the constant (cid:12)(cid:12)(cid:12)(cid:12) m + lp ′ (cid:12)(cid:12)(cid:12)(cid:12) p is optimal.In this section we generalize (5.2) and prove a sharp Hardy inequality with additionalterm and an optimal constant for star-shaped domains and m + l > We start with the following theorem:
Theorem 5.1.
Suppose
Ω = {| x | < ϕ ( x ) } ⊂ R n , n ≥ is a star-shaped domain withrespect to a small ball centered at the origin, p > , m = p − np − , − m < l ≤ − m . Thenfor every u ∈ W ,p ( | x | l (1 − p ) , Ω) , the improved Hardy inequality (cid:18)Z Ω | x | l (1 − p ) (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx (cid:19) p ≥ (cid:18) m + lp ′ (cid:19) (cid:18)Z Ω | u | p | x | n − m − l | ϕ m + l ( x ) − | x | m + l | p dx (cid:19) p + 1 p lim sup ε → ε − n Z S ε | u ( x ) | p dSϕ ( m + l )( p − ( x ) × (cid:18)Z Ω | u | p | x | n − m − l | ϕ m + l ( x ) − | x | m + l | p dx (cid:19) − p ′ , (5.3) holds, where S ε = {| x | = ε } . In inequality (5.3) instead of the distance to zero in the denominator there is thedistance to the boundary on the ray, and the constant is optimal.
Remark 5.1.
For m >
0, i.e., p > n , the choice l = 0 is possible in (5.3) and in this caseHardy inequality (5.3) is true for every u ∈ W ,p (Ω).In order to prove Theorem 5.1 we need some auxiliary results.Fix r < inf | x | =1 ϕ ( x ) and for the annulus A [ r, ϕ ) = { r ≤ | x | < ϕ ( x ) } we introduce thespace W ,p ( A [ r, ϕ )) which is the completion in the norm (5.1) for A [ r, ϕ ) = Ω \ ¯ B r of the C ∞ ( A [ r, ϕ )) functions which are zero in a neighborhood of the boundary S ϕ = {| x | = ϕ ( x ) } (see Maz’ja [80], Ch. 1.1.15 and Ch. 1.1.6), i.e., functions in C ∞ (Ω). The mainelement of the proof of Theorem 5.1 is the following Proposition 5.1 and Corollary 2.2.60 roposition 5.1. Suppose n ≥ , p > , p + 1 p ′ = 1 , − m < l ≤ − m , m = p − np − and f ( x ) = ( f , . . . , f n ) = − x | x | − n (cid:18) ϕ m + l ( x ) − | x | m + l m + l (cid:19) − p , (5.4) where f , f j ∈ C ( A [ r, ϕ )) . Then f satisfies the identity − div f − ( p − | f | p ′ | x | l = 0 , in A [ r, ϕ ) , (5.5) and for every u ∈ W ,p ( | x | l (1 − p ) , A [ r, ϕ )) the inequality Z A [ r,ϕ ) | x | l (1 − p ) (cid:12)(cid:12)(cid:12)(cid:12) h f, ∇ u i| f | (cid:12)(cid:12)(cid:12)(cid:12) p dx ! /p ≥ p ′ Z A [ r,ϕ ) | x | l | f | p ′ | u | p dx ! /p − p Z S r h f, x i| x | | u | p dS Z A [ r,ϕ ) | x | l | f | p ′ | u | p dx ! − p ′ , (5.6) holds.Proof. Without loss of generality we suppose that u ∈ C ∞ (Ω). Simple computations giveus | f | p ′ = | x | m − n (cid:18) ϕ m + l ( x ) − | x | m + l m + l (cid:19) − p , − div f = | x | − n * x, ∇ (cid:18) ϕ m + l ( x ) − | x | m + l m + l (cid:19) − p + = − ( p − | x | − n (cid:18) ϕ m + l ( x ) − | x | m + l m + l (cid:19) − p h ϕ m + l − ( x ) h x, ∇ ϕ ( x ) i − | x | m + l i = ( p − | x | m + l − n (cid:18) ϕ m + l ( x ) − | x | m + l m + l (cid:19) − p = ( p − | f | p ′ | x | l , because h x, ∇ ϕ ( x ) i = 0.Thus we have that (5.5) is satisfied. Inequality (5.6) follows from (2.8) in Corollary2.2 for v = | x | l and w = 0. Proposition 5.2.
Suppose n ≥ , p > , and m = p − np − , − m < l ≤ − m . Then for very u ∈ W ,p ( | x | l (1 − p ) , A [ r, ϕ )) the following inequalities hold: Z A [ r,ϕ ) | x | l (1 − p ) (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx ! p ≥ m + lp ′ Z A [ r,ϕ ) | u | p | x | n − m − l ( ϕ m + l ( x ) − | x | m + l ) p dx ! p + r − n p Z S r | u | p ( ϕ m + l ( x ) − r m + l ) p − dS × Z A [ r,ϕ ) | u | p | x | n − m − l ( ϕ m + l ( x ) − | x | m + l ) p dx ! − p ′ , (5.7) and Z A [ r,ϕ ) | x | l (1 − p ) (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx ≥ (cid:18) m + lp ′ (cid:19) p Z A [ r,ϕ ) | u | p | x | n − m − l ( ϕ m + l ( x ) − | x | m + l ) p dx + r − n p (cid:18) p ′ (cid:19) p − Z S r | u | p ( ϕ m + l ( x ) − r m + l ) p − dS. (5.8) For the functions u k ( x ) = (cid:16) ϕ m + l ( x ) − | x | m + l (cid:17) k Φ( x ) , k > p ′ , (5.9) where Φ( x ) is a homogeneous function of the 0-th order, inequality (5.7) becomes an equal-ity.Proof. Without loss of generality we suppose that u ( x ) is C ∞ function which is zero nearthe boundary S ϕ = {| x | = ϕ ( x ) } . Let us choose f as in Proposition 5.1, see (5.4). Theproof of (5.7) follows from (5.6) with the special choice (5.4) of f since K ( u ) = − Z S r h f, x i| x | | u | p dS = ( m + l ) p − r − n Z S r | u | p | ϕ m + l ( x ) − r m + l | p − dx ≥ . For the proof of (5.8) we use (2.10).For function u k ( x ) in (5.9) we get an equality in (5.7). The proof is similar to theproof of Theorem 5.2 and we omit it. Proof of Theorem 5.1.
Let 0 < ε < inf | x | =1 ϕ ( x ) be a small positive number and u ∈ W ,p ( | x | l (1 − p ) , Ω). Then u ∈ W ,p ( | x | l (1 − p ) , A [ ε, ϕ )), where A [ ε, ϕ ) = { ε ≤ | x | < ϕ ( x ) } .From Corollary 2.2 we get the following Hardy inequality in the annulus A [ ε, ϕ ) for62 m < l ≤ − m , v = | x | l , w = 0 and L ( u ) , K ( u ) , K ( u ) , N ( u ) defined in (2.3) forΩ = A [ ε, ϕ ), i.e., L p ( u ) = Z A [ ε,ϕ ) | x | l (1 − p ) (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx ! p ≥ p ′ K p ( u ) + 1 p K ( u ) K − pp ( u )= m + lp ′ Z A [ ε,ϕ ) | u | p | x | n − m − l | ϕ m + l ( x ) − | x | m + l | p dx ! p + ε − n p Z S ε | u | p | ϕ m + l ( x ) − ε m + l | p − dS × Z A [ ε,ϕ ) | u | p | x | n − m − l | ϕ m + l ( x ) − | x | m + l | p dx ! − pp . (5.10)After the limit ε → Z S ε | u | p | ϕ m + l ( x ) − ε m + l | p − dS ≥ Z S ε | u | p ϕ ( m + l )( p − ( x ) dS, we get (5.3). The inequality (5.3) becomes an equality for a class of functions defined in Theorem 5.2.Moreover, in Corollary 5.2 we show that in the special case of a ball, inequality (5.3)transforms into (5.21) with the distance function in the denominator and the optimalconstant.
Theorem 5.2.
Suppose
Ω = {| x | < ϕ ( x ) } ⊂ R n , n ≥ is a star-shaped domain withrespect to a small ball centered at the origin p > , m = p − np − , − m < l ≤ − m . ThenHardy inequality (5.3) is an equality if u ( x ) = u k ( x ) , u k ( x ) = (cid:16) ϕ m + l ( x ) − | x | m + l (cid:17) k Φ( x ) , k > p ′ , where Φ is a homogeneous function of the 0–th order. Moreover, the constant (cid:18) m + lp ′ (cid:19) p is optimal for (5.2) and (5.3).Proof. From Theorem 2.2 it follows that (5.3) is an equality if and only if for a.e. x ∈ Ωand some constant k ≥ x ∈ Ω. u h f, ∇ u i = | u h f, ∇ u i| , h f, ∇ u i = k v | f | p ′ u. (5.11)63rom the choice of f in (5.4) equalities (5.11) are equivalent to − u h x, ∇ u i = | u h x, ∇ u i| , h x, ∇ u i = − k | x | m + l (cid:16) ϕ m + l ( x ) − | x | m + l (cid:17) − u, (5.12)for a.e. x ∈ Ω.We are looking for solution of the second equation in (5.12) of the form u = (cid:16) ϕ m + l ( x ) − | x | m + l (cid:17) k m + l z. Together with (5.12) a simple computation gives us − k | x | m + l (cid:16) ϕ m + l ( x ) − | x | m + l (cid:17) − u = h x, ∇ u i = k (cid:16) ϕ m + l ( x ) − | x | m + l (cid:17) k m + l − (cid:16) ϕ m + l − ( x ) h x, ∇ ϕ ( x ) i − | x | m + l − h x, x i (cid:17) z + (cid:16) ϕ m + l ( x ) − | x | m + l (cid:17) k m + l h x, ∇ z i = − k | x | m + l (cid:16) ϕ m + l ( x ) − | x | m + l (cid:17) k m + l − z + (cid:16) ϕ m + l ( x ) − | x | m + l (cid:17) k m + l h x, ∇ z i = − k | x | m + l (cid:16) ϕ m + l ( x ) − | x | m + l (cid:17) − u + (cid:16) ϕ m + l ( x ) − | x | m + l (cid:17) k m + l h x, ∇ z i . Hence z is a solution of the homogeneous first order partial differential equation h x, ∇ z i = 0 , in Ω \{ } . (5.13)It is well known, see Evans [32], that all solutions of (5.13) are in the form of z ( x ) = Φ( x ),where Φ is a homogeneous function of the 0-th order, i.e., all solutions of (5.12) are u k = (cid:16) ϕ m + l ( x ) − | x | m + l (cid:17) k Φ( x ) for k ≥
0. When k > p ′ , then u k ∈ W ,p ( | x | l (1 − p ) , Ω).Let us check up that for u k inequality (5.3) becomes an equality. Since ϕ ( x ) and Φ( x )are homogeneous functions of the 0-th order we have h x, ∇ ϕ ( x ) i = 0, h x, ∇ Φ( x ) i = 0.We make a polar change of the variables and for simplicity we use the same notations ϕ ( x ) , Φ( x ) for this functions depending only on the angular variables. We obtain thefollowing chain of equalities for the left-hand side of (5.3): L ( u k ) = Z Ω | x | l (1 − p ) (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u k i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx = [ k ( m + l )] p Z Ω | Φ( x ) | p | x | l (1 − p )+ p ( m + l − (cid:12)(cid:12)(cid:12) ϕ m + l ( x ) − | x | m + l (cid:12)(cid:12)(cid:12) p ( k − dx = [ k ( m + l )] p Z S Z ϕ ( x )0 | Φ( x ) | p ρ m + l − | ϕ m + l ( x ) − ρ m + l | p ( k − dρdS = k p ( m + l ) p − Z S Z ϕ ( x )0 | Φ( x ) | p | ϕ m + l ( x ) − ρ m + l | p ( k − dρ m + l dS = k p ( m + l ) p − pk − p + 1 Z S | Φ( x ) | p ϕ ( m + l )( pk − p +1) ( x ) dS < ∞ , (5.14)64ecause pk − p + 1 > k > p ′ and m + l > K ( u k ), K ( u k ) in the right-hand side of (5.3) we obtain K ( u k ) = Z Ω | u k | p | x | n − m − l | ϕ m + l ( x ) − | x | m + l | p dx = Z S Z ϕ | Φ( x ) | p ρ m + l − | ϕ m + l ( x ) − ρ m + l | p − pk dρdS = 1( m + l )( pk − p + 1) Z S | Φ( x ) | p ϕ ( m + l )( pk − p +1) ( x ) dS, (5.15)and K ( u k ) = 1 p lim ε → ε − n Z S ε | u k | p | ϕ m + l ( x ) − ε m + l | p − dS = 1 p Z S | Φ( x ) | p ϕ ( m + l )( pk − p +1) dS. Thus for the right-hand side of (5.3) we get finally K ( u k ) = (cid:18) m + lp ′ K ( u k ) + K ( u k ) (cid:19) K − p ′ ( u k )= (cid:20) ( m + l )( p − m + l )( pk − p + 1) p + 1 p (cid:21) [( m + l )( pk − p + 1)] p ′ × (cid:20)Z S | Φ( x ) | p ϕ ( m + l )( pk − p +1) dS (cid:21) p = ( m + l ) p ′ k ( pk − p + 1) p (cid:20)Z S | Φ( x ) | p ϕ ( m + l )( pk − p +1) dS (cid:21) p . So the left-hand side ( L ( u k )) /p of (5.3) coincides with the right-hand side K ( u k ) of(5.3). Thus (5.3) is an equality for u k ( x ).Let us check now that the constant (cid:18) m + lp ′ (cid:19) p is optimal for (5.3). From (5.3), (5.14)and (5.15) we have (cid:18) m + lp ′ (cid:19) p ≤ Z Ω | x | l (1 − p ) (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u k i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx × (cid:18)Z Ω | u k | p | x | n − m − l | ϕ m + l ( x ) − | x | m + l | p dx (cid:19) − = ( m + l )( pk − p + 1) k p ( m + l ) p − ( pk − p + 1)= (cid:18) m + lp ′ (cid:19) p ( p ′ k ) p → (cid:18) m + lp ′ (cid:19) p + 0 , when k → p ′ + 0 . B R = {| x | < R } and l = 0. Corollary 5.1.
Suppose B R = {| x | < R } ⊂ R n , p > n ≥ , m = p − np − . Then for every u ∈ W ,p ( B R ) Hardy inequality (cid:18)Z B R (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx (cid:19) p ≥ p − np (cid:18)Z B R | u | p | x | n − m | R m − | x | m | p dx (cid:19) p + 1 p R n − p ω n | u (0) | p (cid:18)Z B R | u | p | x | n − m | R m − | x | m | p dx (cid:19) − p ′ , (5.16) holds, where ω n is the (n-1)-dimensional measure of the unite sphere S .For u k ( x ) = ( R m − | x | m ) k Φ( x ) , k > p ′ , where Φ is a homogeneous function of the 0–th order, (5.16) becomes an equality, i.e.,(5.16) is sharp and the constant p − np is optimal.Proof. Corollary 5.1 follows from Theorem 5.1 for ϕ ( x ) = R , the constant p − np in (5.16)is optimal and inequality (5.16) is sharp due to Theorem 5.2.As it is shown in Sect. 1.2, different forms of Hardy inequalities with additional termand optimal constant are considered in Brezis and Marcus [25], Hoffmann-Ostenhof et al.[59], Tidblom [94], Filippas et al. [44], Avkhadiev and Wirths [11], etc. In Hardy inequality(5.16), the weights in the leading term of the right-hand side have singularities at 0 andon the boundary of the ball B R . This inequality is not in the ‘linear’ form and it is sharp.Moreover, in Corollary 5.2 we obtain the inequality (5.21) where the leading term in theright-hand side is written with the distance function d ( x ) and an additional term whichdepends on the value of the function at 0.Inequalities (5.3) and (5.16) are sharp, but they are not in a ‘linear’ form. Using Younginequality, we can get a ‘linear’ form of these inequalities. Theorem 5.3.
Suppose
Ω = {| x | < ϕ ( x ) } ⊂ R n , n ≥ is a star-shaped domain withrespect to a small ball centered at the origin, p > , m = p − np − , − m < l ≤ − m . Thenthe following Hardy inequality holds in Ω : Z Ω | x | l (1 − p ) (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx ≥ (cid:18) m + lp ′ (cid:19) p Z Ω | u | p | x | n − m − l | ϕ m + l ( x ) − | x | m + l | p dx + (cid:18) m + lp ′ (cid:19) p − p lim sup ε → ε − n Z S ε | u ( x ) | p dSϕ ( m + l )( p − ( x ) , u ∈ W ,p ( | x | l (1 − p ) , Ω) , When p > n, l = 0 then in B R the inequality Z B R (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx ≥ (cid:18) p − np (cid:19) p Z B R | u | p | x | n − m | R m − | x | m | p dx + (cid:18) p − np (cid:19) p − R n − p p ω n | u (0) | p , u ∈ W ,p ( B R ) , (5.17)66 olds.Proof. Since K ( u ) ≥
0, we have from (2.10) that L ( u ) ≥ (cid:18) p ′ (cid:19) p K ( u ) + (cid:18) p ′ (cid:19) p − K ( u ) . (5.18)The rest of the proof follows from (5.18), Theorem 5.1 and Corollary 5.1.As a corollary of Theorems 5.1 and 5.3 we get Corollary 5.2. If m = p − np − > , then (cid:18) p − np (cid:19) p Z B R | u | p | x | n − m | R m − | x | m | p dx ≥ (cid:18) p ′ (cid:19) p Z B R | u | p d p ( x ) dx (5.19) and correspondingly (cid:18)Z B R (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx (cid:19) p ≥ p ′ (cid:18)Z B R | u | p d p ( x ) dx (cid:19) p + R n − p p ω n | u (0) | p (cid:18)Z B R | u | p | x | n − m | R m − | x | m | p dx (cid:19) − p ′ , (5.20) Z B R (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx ≥ (cid:18) p ′ (cid:19) p Z B R | u | p d p ( x ) dx + (cid:18) p − np (cid:19) p − R n − p p ω n | u (0) | p , (5.21) hold for every u ∈ W ,p ( B R ) . Moreover, the constant (cid:18) p ′ (cid:19) p in (5.20) is optimal.Proof. It is enough to prove the inequality (cid:18) p − np (cid:19) p | x | m − n ( R m − | x | m ) − p ≥ (cid:18) p − p (cid:19) p ( R − | x | ) − p , or equivalently ( p − n )( R − ρ ) ≥ ( p − ρ n − p − ( R m − ρ m ) , (5.22)for | x | = ρ and n − m = ( n − pp − h ( ρ ) = ( p − n )( R − ρ ) − ( p − ρ n − p − ( R m − ρ m )= ( p − n )( R − ρ ) − ( p − (cid:16) R m ρ n − p − − ρ (cid:17) , is a decreasing one for ρ ∈ [0 , R ] because h ′ ( ρ ) = ( n − p ) − ( n − R m ρ n − p − − + p −
1= ( n − (cid:20) − (cid:18) Rρ (cid:19) m (cid:21) ≤ , and m >
0. Since h ( R ) = 0 inequality (5.22) is satisfied.To obtain inequalities (5.20) and (5.21) we replace (5.19) in (5.16) and (5.17). Theoptimality of the constants 1 p ′ in (5.20) and (cid:18) p ′ (cid:19) p in (5.21) follows from Corollary 5.1.67 Estimates from below for the first eigenvalue of the p-Laplacian
In this section we give an application of Hardy inequalities for the estimate from below ofthe first eigenvalue λ p,n of the p–Laplacian ∆ p u = div( |∇ u | p − ∇ u ), p > ⊂ R n , n ≥ ∂ Ω − ∆ p u = λ p,n | u | p − u in Ω ,u = 0 on ∂ Ω . (6.1) Here we listed some estimates of λ p,n . The first eigenvalue λ p,n (Ω) can be characterizedthrough Reyleigh quotient, see Cheeger [27], Lindqvist [77] λ p,n (Ω) = inf u ∈ W ,p (Ω) Z Ω |∇ u | p dx Z Ω | u | p dx , (6.2)and λ p,n (Ω) is simple, i.e., the first eigenfunction u p,n ( x ) is unique up to multiplicationwith non-zero constant C . Moreover, u p,n ( x ) is positive in Ω, u p,n ( x ) ∈ W ,p (Ω) ∩ C ,δ ( ¯Ω)for some δ ∈ (0 ,
1) (see for example Belloni and Kawohl [17] and the references therein).Analytical values of λ p,n are known only for p > n = 1 or p = 2 and n ≥
2. For p > n = 1, Ω = ( a, b ), see ˆOtani [85], the analytical value of λ p, (Ω) is λ p, (Ω) = ( p − π ( b − a ) p sin πp ! p . For n ≥ p = 2, i.e., for the Laplace operator, the value of λ ,n (Ω) is knownby analytical formulae for domains Ω with simple geometry like a ball, a spherical shell, aparallelepiped etc. Numerical approximations have been done for more general domains,see Vladimirov [96] and the review by Grebenkov and Nguyen [53]. For example, if Ω is aball centered at zero, B R ⊂ R n then λ ,n ( B R ) = µ ( α )1 R ! , α = n − , where µ ( α )1 is the first positive zero of the Bessel function J α .If p = 2, the explicit value of λ p,n (Ω) is not known even for domains Ω like a ball or acube. That is why an explicit lower bound for λ p,n (Ω) is an important task.For this purpose the Faber–Krahn theorem simplifies the estimate of the first eigenvaluefor arbitrary domain to the estimate in a ball. Let us recall the Faber–Krahn inequalitywhich gives an estimate from below of λ p,n (Ω) for arbitrary bounded domain Ω ⊂ R n with λ p,n (Ω ∗ ), where Ω ∗ is the n-dimensional ball of the same volume as Ω, see Lindqvist[77], Bhattacharia [21], Huang [60], Kawohl and Fridman [63]. In Kawohl and Fridman[63] is proved that among all domains Ω of a given n-dimensional volume the ball Ω ∗ withthe same volume as Ω minimizes λ p,n (Ω), in other words λ p,n (Ω) ≥ λ p,n (Ω ∗ ) . stimate with a Cheeger’s constant One of the first lower bounds for λ p,n (Ω) is based on Cheeger’s constant h (Ω) = inf D ⊂ Ω | ∂D || D | . Here D varies over all smooth sub-domains of Ω whose boundary ∂D does not touch ∂ Ω,where | ∂D | and | D | are the (n-1)- and n-dimensional Lebesgue measure of ∂D and D respectively.In Cheeger [27] for p = 2 and in Lefton and Wei [71] for p > λ p,n (Ω) ≥ (cid:18) h (Ω) p (cid:19) p . (6.3)Inequality (6.3) is sharp for p →
1, because λ p,n (Ω) converges to the Cheeger’s constant h (Ω), see Kawohl and Fridman [63], Corollary 6.The Cheeger’s constant h (Ω) is known only for special domains. For example, if Ω isa ball B R ⊂ R n , then h (Ω) = nR and (6.3) gives the following lower bound for λ p,n ( B R ),see Kawohl and Fridman [63], λ p,n ( B R ) ≥ Λ (1) p,n ( B R ) = (cid:18) npR (cid:19) p , for p > , n ≥ . (6.4)Thus combining the above results the following inequality holds for p →
1, see Kawohland Fridman [63], Remark 5, λ ,n (Ω) ≥ n (cid:18) ω n | Ω | (cid:19) /n = Λ (1)1 ,n (Ω) , n ≥ , (6.5)where ω n is the volume of the unit ball in R n . If Ω is a ball, then (6.5) becomes an equality.In the other limit case p → ∞ the result in Juutinen et al. [62] says that λ ∞ ,n (Ω) = lim p →∞ ( λ p,n (Ω)) /p = max { dist( x, ∂ Ω) , x ∈ Ω } − . In particular for Ω = B R λ ∞ ,n ( B R ) = lim p →∞ (cid:16) Λ (1) p,n ( B R ) (cid:17) /p = 1 R .
Estimate with Picone’s identity
In Benedikt and Dr´abek [18], Theorem 2, and in Benedikt and Dr´abek [19], Theorem 2,by Picone’s identity the following estimate for p > λ p,n ( B R ) ≥ Λ (2 , p,n ( B R ) = nR p (cid:18) pp − (cid:19) p − , Λ (2 , p,n ( B R ) = npR p . n Λ (2 , p,n ( B R ) , Λ (2 , p,n ( B R ) o = ( Λ (2 , p,n ( B R ) , for 1 < p < , Λ (2 , p,n ( B R ) , for p ≥ , see Proposition 6.4, the estimate λ p,n ( B R ) ≥ Λ (2) p,n ( B R ) = Λ (2 , p,n ( B R ) = nR p (cid:18) pp − (cid:19) p − , for 1 < p < , Λ (2 , p,n ( B R ) = npR p , for p ≥ , (6.6)holds. Estimate with Sobolev constant
It is not difficult to estimate λ p,n (Ω) from below in a bounded domain for 1 < p < n bythe well-known Sobolev and H¨older inequalities k∇ u k p ≥ C n,p k u k npn − p ≥ C n,p k u k p | Ω | − /n . (6.7)The best Sobolev’s constant C n,p is obtained in Aubin [8] and Talenti [93]. For moredetails see Maz’ja [80] and Ludwig et al. [79] C n,p = n /p ω /nn (cid:18) n − pp − (cid:19) p − p Γ (cid:0) n (cid:1) Γ (cid:16) n + 1 − np (cid:17) Γ( n ) /n . From (6.7) the estimate from below of the first eigenvalue for Ω = B R becomes λ p,n ( B R ) ≥ nR p (cid:18) n − pp − (cid:19) p − Γ (cid:0) n (cid:1) Γ (cid:16) n + 1 − np (cid:17) Γ( n ) p/n = Λ ( S ) p,n ( B R ) , for 1 < p < n. Lindqvist’s estimate in parallelepiped
In the parallelepiped P = { x ∈ R n , < x j < a j , j = 1 , , . . . , n } with a min = min i ≤ i ≤ n a i for p > n we have the estimate λ p,n ( P ) ≥ pa pmin , by Lindqvist [78]. If Ω is an arbitrary bounded domain in R n and R is the radius ofthe largest ball inscribed in the smallest parallelepiped (with minimal a min ) containing Ω,then λ p,n (Ω) ≥ pR p = Λ ( L ) p,n ( B R ) , for p > n. (6.8)70 umerical estimates In Biezuner et al. [22, 23], different numerical methods for computing λ p,n (Ω) inspiredby the inverse power method in finite dimensional algebra are developed. By means ofiterative technique the authors define two sequences of functions. One of the sequences ismonotone decreasing, the other one is monotone increasing. The first eigenvalue λ p,n (Ω)is between the limits of these sequences. In the case of a ball the two limits are equal and λ p,n (Ω) coincides with them.In Lefton and Wei [71] a finite element technique for numerical approximation of thefirst eigenfunction and the first eigenvalue of (6.1) is used. λ p,n using Hardy inequalities We prove several analytical bounds from below of the first eigenvalue of the p–Laplacianin bounded domains using different Hardy inequalities with weights derived in section 2.The section is based on the results in Fabricant et al. [36, 38, 39], Kutev and Rangelov[68, 69].For this purpose we use the following Faber–Krahn theorem.
Theorem 6.1 (Kawohl and Fridman [63]) . Among all domains of a given n-dimensionalvolume the ball Ω ∗ with the same volume as Ω minimizes every λ p,n (Ω) , in other words λ p,n (Ω) ≥ λ p,n (Ω ∗ ) . (6.9)Thus, from (6.9) it is enough for us to find an estimate for λ p,n only in the ball Ω ∗ = B R .Hardy inequalities are with weights singular either at some interior point of Ω, usuallyat the origin, or with weights singular on the boundary or combined double singularitiesat 0 and at ∂ Ω. We will apply these three types of Hardy inequalities in order to estimatefrom below the first eigenvalue λ p,n (Ω). We are concentrating only on those Hardy’sinequalities (among the large number of results in the literature) which are with explicitlygiven constants.Let us note that from the classical Hardy inequality, see (1.3) using the Reyleighquotient (6.2), we get immediately the estimate λ p,n ( B R ) ≥ (cid:12)(cid:12)(cid:12)(cid:12) n − ppR (cid:12)(cid:12)(cid:12)(cid:12) p = Λ ( H ) p,n ( B R ) , for n ≥ , p > , n = p. (6.10) Estimates by means of Hardy inequalities with double singular weights
From (2.37), (2.38) and (2.39) for u ∈ W ,p ( B R ) ignoring the boundary terms we haveHardy inequalities Z B R |∇ u | p dx ≥ (cid:12)(cid:12)(cid:12)(cid:12) p − np (cid:12)(cid:12)(cid:12)(cid:12) p Z B R | u | p | x | n − m | R m − | x | m | p dx, for p = n, (6.11) Z B R |∇ u | n dx ≥ (cid:18) n − n (cid:19) n Z B R | u | n | x | n (cid:12)(cid:12)(cid:12) ln R | x | (cid:12)(cid:12)(cid:12) n dx, for p = n, (6.12)where p > n ≥ m = p − np − λ p,n ( B R ).71 heorem 6.2. For every n ≥ , p > the following estimates hold:(i) If p = n , then λ p,n ( B R ) ≥ (cid:18) pR (cid:19) p (cid:20) ( n − n − ( p − p − (cid:21) pn − p . (6.13) (ii) If p=n, then λ n,n ( B R ) ≥ (cid:18) n − nR (cid:19) n e n . (6.14) Proof. (i) If | x | = ρ ∈ [0 , R ), then for every x ∈ B R and p = n we get for the right-handside of (6.11) the estimate Z B R | u | p | x | n − m | R m − | x | m | p dx ≥ inf ρ ∈ (0 ,R ) (cid:0) ρ n − m | R m − ρ m | p (cid:1) − Z B R | u | p dx. Further on we will use the identities m − n = ( m − p = (1 − n ) pp − < , − m = n − p − > ,n − mm = (1 − m ) pm = ( n − pp − n . (6.15)Now applying the definition of λ p,n ( B R ) with Reyleigh quotient from (6.2) and (6.15)we have λ p,n ( B R ) ≥ (cid:12)(cid:12)(cid:12)(cid:12) p − np (cid:12)(cid:12)(cid:12)(cid:12) p inf ρ ∈ (0 ,R ) (cid:2) ρ − m | R m − ρ m | (cid:3) − p = (cid:12)(cid:12)(cid:12)(cid:12) p − np (cid:12)(cid:12)(cid:12)(cid:12) p " sup ρ ∈ (0 ,R ) (cid:0) ρ − m | R m − ρ m | (cid:1) − p . (6.16)For the function z ( ρ ) = ρ − m | R m − ρ m | in the interval (0 , R ) we have z ′ ( ρ ) = (cid:2) (1 − m ) R m ρ − m − (cid:3) sgn( m ) , and z ′ ( ρ ) = 0 only at the point ρ = R (1 − m ) /m = R (cid:18) n − p − (cid:19) /m . For m >
0, i.e., p > n we have n − p − < (cid:18) n − p − (cid:19) m < m <
0, i.e., p < n theinequality n − p − > m < (cid:18) n − p − (cid:19) m < < (cid:18) n − p − (cid:19) /m < m = 0, then 0 < ρ < R and from z ′′ ( ρ ) = −| m | (1 − m ) R m ρ − m − < z ( ρ ) has a maximumat the point ρ and z ( ρ ) = R (cid:18) n − p − (cid:19) n − p − n (cid:12)(cid:12)(cid:12)(cid:12) p − np − (cid:12)(cid:12)(cid:12)(cid:12) . (6.17)72ence from (6.16) and (6.17) we get λ p ( B R ) ≥ (cid:18) p − p (cid:19) p (cid:18) n − p − (cid:19) − ( n − pp − n R − p = (cid:18) Rp (cid:19) p (cid:20) ( n − n − ( p − p − (cid:21) pn − p . (ii) As in the proof of (6.13) for the right-hand side of (6.12) we get the estimate Z B R | u | n | x | n (cid:12)(cid:12)(cid:12) ln R | x | (cid:12)(cid:12)(cid:12) n dx ≥ inf ρ ∈ (0 ,R ) (cid:18) ρ ln Rρ (cid:19) − n Z B R | u | n dx. From (6.12) and (6.2) we obtain λ n,n ( B R ) ≥ (cid:18) n − n (cid:19) n inf ρ ∈ (0 ,R ) (cid:18) ρ ln Rρ (cid:19) − n = (cid:18) n − n (cid:19) n " sup ρ ∈ (0 ,R ) ρ ln Rρ − n . (6.18)For the function y ( ρ ) = ρ ln Rρ in the interval (0 , R ) we have y ′ ( ρ ) = ln Rρ − y ′ ( ρ ) = 0 only at the point ρ = Re − ∈ (0 , R ). Since y ′′ ( ρ ) = − ρ <
0, the function y ( ρ ) has a maximum at ρ and y ( ρ ) = Re − .Hence from (6.18) we obtain λ n,n ( B R ) ≥ (cid:18) n − n (cid:19) n (cid:0) Rr − (cid:1) − n = (cid:18) n − Rn (cid:19) n e n . Estimates by means of Hardy inequalities with additional logarithmic term
In this section we estimate from below the first eigenvalue of the p–Laplacian in B R ⊂ R n , n ≥ p > n , m = p − np − > Z B R |∇ u | p dx ≥ Z B R (cid:12)(cid:12)(cid:12)(cid:12) h x, ∇ u i| x | (cid:12)(cid:12)(cid:12)(cid:12) p dx ≥ (cid:18) p − np (cid:19) p Z B R " p p −
1) 1ln R m −| x | m eτ R m | u | p | x | ( n − p ′ | R m − | x | m | p dx (6.19)where τ = e y − and y = 1 − p | a |− | a | = 2 s p − p − ! − for a = − p − p − . (6.20)73 heorem 6.3. For every ball B R ∈ R n , n ≥ , p > n , m = p − np − the estimate λ p,n ( B R ) ≥ (cid:18) pR (cid:19) p (cid:20) ( p − p − ( n − n − (cid:21) pp − n × p p − " s p − p − − m − τ − , (6.21) holds, where τ = 1 − − m ) " p s p − p − − m ! − m − ∈ (0 , . (6.22) Proof.
Suppose that Z B R | u | p dx = 1 and with the notation ε = 1 eτ , i.e.,ln ε = − y = − s p − p − ! , (6.23)from (6.20) and from (6.19) we obtain the estimate λ p,n ( B R ) ≥ (cid:18) p − npR (cid:19) p inf ρ ∈ [0 ,R ] ( (cid:0) ρR (cid:1) n − m (cid:0) − (cid:0) ρR (cid:1) m (cid:1) p × " p p −
1) 1ln ε (cid:0) − (cid:0) ρR (cid:1) m (cid:1) = (cid:18) p − npR (cid:19) p inf z ∈ [0 , ((cid:20) p p −
1) 1ln εz (cid:21) z p (1 − z ) n − mm ) ≥ (cid:18) p − npR (cid:19) p ( inf z ∈ [0 , z p (1 − z ) n − mm + inf z ∈ [0 , p p −
1) 1ln εz z p (1 − z ) n − mm ) = (cid:18) p − npR (cid:19) p (cid:18) I + p p − I (cid:19) , (6.24)where I = inf z ∈ [0 , z p (1 − z ) n − mm = " sup z ∈ [0 , z p (1 − z ) n − mm − ,I = inf z ∈ [0 , z p (1 − z ) n − mm ln εz = " sup z ∈ [0 , z p/ (1 − z ) n − m m ( − ln εz ) − . (6.25)74ince n − mm > h ( z ) = z p (1 − z ) n − mm satisfies the conditions h (0) = h (1) = 0 and h ′ ( z ) = pz p − (1 − z ) n − mm − n − mm z p (1 − z ) n − mm − = pz p − (1 − z ) n − mm − (cid:16) − zm (cid:17) , so it follows that h ( z ) has a maximum at the point m ∈ (0 , z ∈ [0 , h ( z ) = h ( m ) = m p (1 − m ) n − mm = (cid:18) p − np − (cid:19) p (cid:18) n − p − (cid:19) p ( n − p − n , and we get (cid:18) p − npR (cid:19) p I = (cid:18) p − npR (cid:19) p (cid:18) p − np − (cid:19) − p (cid:18) n − p − (cid:19) − p ( n − p − n = (cid:18) pR (cid:19) p (cid:18) ( n − n − ( p − p − (cid:19) pn − p which gives the same estimate from below for λ p,n ( B R ) as in (6.13).Let us estimate I . For the function G ( z ) = z p (1 − z ) n − m m ( − ln εz ) we have G ′ ( z ) = z p − (1 − z ) n − m m − g ( z ),where g ( z ) = p − z )( − ln εz ) − z − n − m m z ( − ln εz )= 12 (cid:20) p − (cid:18) p + p ( n − p − n (cid:19)(cid:21) ( − ln εz ) − z = p (cid:16) − zm (cid:17) ( − ln εz ) − z. Simple computations give us g ′ ( z ) = p (cid:18) m − z + 1 m ln εz (cid:19) + 1, g ′′ ( z ) = p (cid:18) mz + 1 z (cid:19) >
0, for z ∈ [0 , g ′ ( z ), (6.15) and (6.23) we get the chain of equalitiessup z ∈ [0 , g ′ ( z ) = g ′ (1) = p m (1 − m + ln ε ) + 1= p m n − p − − − s p − p − ! + 1= p p − n ) " − p − n ) − p p − r p − − p p − ! + 1 < − p ( p − n )4( p − n ) + 1 < − p < , for p > n ≥ . g ′ ( z ) < z ∈ [0 ,
1] and lim z → g ( z ) = ∞ , g ( m ) = m − <
0, it follows that thereexists a unique point z ∗ ∈ (0 , m ) such that g ( z ∗ ) = 0, i.e., G ′ ( z ∗ ) > z ∈ [0 , z ∗ ), G ′ ( z ) < z ∈ ( z ∗ , z ∈ [0 , G ( z ) = sup z ∈ (0 ,m ] G ( z ) = G ( z ∗ ) . (6.26)In order to localize better the maximum point z ∗ we look for z = τ m , τ ∈ (0 ,
1] such that G ′ ( τ m ) >
0. From (6.23) we get the chain of equalities g ( τ m ) = p − τ ) ( − ln( ετ m )) − τ m = p − τ ) ( − ln( τ )) + p s p − p − − m ! (1 − τ ) − τ m = p − τ ) ( − ln( τ )) + p s p − p − − m ! − − τ " − m + p s p − p − − m ! = p − τ )( − ln τ ) . (6.27)Since m = p − np − < m < p s p − p − − m ! = p s p − p − − m ! > p > > m, − − m ) " p s p − p − − m ! − m − > − − m )(4 − m ) − = 0 . Hence 0 < τ = 1 − − m ) " p s p − p − − m ! − m − < , (6.28)and from (6.27) the inequality G ′ ( τ m ) > , (6.29)is satisfied. Thus z ∗ ∈ ( τ m, m ), where τ is given in (6.22).76rom (6.25), (6.26), (6.28) and (6.29) it follows that − ln( εz ) ≤ − ln( ετ m ) for every z ∈ [ τ m, m ] and sup z ∈ [ τm,m ] h ( − ln( εz )) z p (1 − z ) n − m m i ≤ ( − ln( ετ m )) sup z ∈ [ τm,m ] z p (1 − z ) n − m m = ( − ln( ετ m )) sup z ∈ [ τm,m ] h ( z ) = ( − ln( ετ m )) h ( m )= ( − ln ετ m ) m p (1 − m ) n − m m = " s p − p − ! − ln τ − ln m I − from the considerations for I . Thus we have the following estimate for I I ≥ I ( − ln ετ m ) − = 12 " s p − p − − m − τ − and hence from (6.24) we obtain (6.21). Estimates by means of one-parametric family of Hardy inequalities
We will obtain a new analytical estimate for λ p,n ( B R ) from below using one-parametricfamily of Hardy inequalities developed in section 2.5.For this purpose we introduce the notations: A ( p, n, δ ) = ( p − p − δ − p ( n − δ )]; B ( p, n, δ ) = ( p − n − δ )( p + δ ) − ( δ − p − δ ); C ( p, n, δ ) = − δ ( n − δ )( p − , D = B − AC ; A ( p, n ) = − p ( n − p ) , B ( p, n ) = p ( p − n − p − C ( p, n ) = p ( p − , D = B − A C . (6.30)Consider the quadratic equations Az + Bz + C = 0 and Cy + By + A = 0 , (6.31) A z + B z + C = 0 , (6.32)77nd note that their discriminants D and D are correspondingly D = B − AC = [( p − n − δ )( p + δ ) − ( δ − p − δ )] + 4( p − p − δ − p ( n − δ )] δ ( n − δ )( p − p − δ ) (cid:8) [( p − n − δ ) + 1 − δ ] + 4 δ ( p − n − δ ) (cid:9) = ( p − δ ) D > , (6.33)for p > , δ ∈ (0 , n ), n ≥ δ = p , D = B − A C = p ( p − p − n − p − + 4 p ( n − p )] > . Let us define some of the roots of (6.31), (6.32) z + ( p, n, δ ) = − B ( p, n, δ ) + p D ( p, n, δ )2 A ( p, n, δ ) ,y + ( p, n, δ ) = − B ( p, n, δ ) + p D ( p, n, δ )2 C ( p, n, δ ) ,z − ( p, n ) = − B ( p, n ) − p D ( p, n )2 A ( p, n ) . (6.34)In this section our main result is: Theorem 6.4.
For n ≥ , p > , if Σ = (cid:26) δ ∈ (0 , n ) , δ = p n − p − (cid:27) then the estimate λ p,n ( B R ) ≥ Λ (3) p,n ( B R ) = 1 R p sup δ ∈ Σ H ( p, n, δ ) , (6.35) holds where H ( p, n, δ )= " p − δp (1 − z + ( p, n, δ )) z + ( p, n, δ ) δp − δ p − (cid:20) ( p − δ ) z + ( p, n, δ ) p (1 − z + ( p, n, δ )) + n − δ (cid:21) , < δ < p, " p − pe − z − ( p,n ) ( z − ( p, n )) p − " ( p − pe − z − ( p,n ) ( z − ( p, n )) + n − pe − z − ( p,n ) , δ = p, " δ − pp (1 − y + ( p, n, δ )) y + ( p, n, δ ) pδ − p p − (cid:20) ( δ − p ) p (1 − y + ( p, n, δ )) + n − δ (cid:21) , p < δ. Let us consider some special cases for δ : • Suppose δ →
0, so that from (6.35) and A ( p, n,
0) = − p ( p − n − B ( p, n,
0) = p [( p − n + 1], C ( p, n,
0) = 0, D ( p, n,
0) = p [( p − n + 1] , we obtain z + ( p, n,
0) = 0.Applying the L’Hospital rule we get 78im δ → H ( p, n, δ ) = n lim δ → z + ( p, n, δ ) − δ ( p − p − δ = n lim δ → e − δ ( p − p − δ ln z + ( p,n,δ ) = n, so that Λ (3) p,n ( B R ) ≥ Λ (3 , p,n ( B R ) = nR p . • We let δ → n in (6.35) and from A ( p, n, n ) = ( p − p − n ), B ( p, n, n ) = ( n − p )( n − C ( p, n, n ) = 0, D ( p, n, n ) =( p − n ) ( n − , A ( n, n ) = 0, B ( n, n ) = − n ( n − C ( n, n ) = n ( n − D ( n, n ) = n ( n − we getlim δ → n z + ( p, n, δ ) = n − p − , − lim δ → n z + ( p, n, δ ) = p − np − , for n < p, lim δ → n y + ( p, n, δ ) = p − n − , − lim δ → n y + ( p, n, δ ) = n − pn − , for p < n. lim p → n z − ( p, n ) = 1 , for p = n. So Λ (3) p,n ( B R ) ≥ Λ (3 , p,n ( B R )= 1 R p lim δ → n H ( p, n, δ ) = (cid:18) pR (cid:19) p (cid:20) ( n − n − ( p − p − (cid:21) pn − p , for p = n. (6.36)andΛ (3) n,n ( B R ) ≥ Λ (3 , n,n ( B R ) = 1 R n lim δ → n H ( δ, n, δ ) = (cid:18) n − nR (cid:19) n e n , for p = n. (6.37)Estimates (6.36), (6.37) coincide with the result in Theorem 6.2, estimates (6.13)and (6.14). Proof of Theorem 6.4.
The proof follows by means of the estimate from below of thekernels of the integrals in the right-hand side of (2.61) and (2.62).We will consider cases δ = p and δ = p separately.(i) For the case δ = p , δ ∈ (0 , n ) we have the following estimate from Hardyinequality (2.61) Z B R |∇ u | p dx ≥ Z B R g ( | x | ) | u | p dx, (6.38)where g ( | x | ) = (cid:12)(cid:12)(cid:12)(cid:12) p − δp (cid:12)(cid:12)(cid:12)(cid:12) p | x | ( δ − pp − (cid:12)(cid:12)(cid:12) R p − δp − − | x | p − δp − (cid:12)(cid:12)(cid:12) p + ( n − δ ) (cid:12)(cid:12)(cid:12)(cid:12) p − δp (cid:12)(cid:12)(cid:12)(cid:12) p − | x | δ (cid:12)(cid:12)(cid:12) R p − δp − − | x | p − δp − (cid:12)(cid:12)(cid:12) p − . | x | = Rr , r ∈ (0 ,
1) we get g ( | x | ) = g ( Rr ) = R − p G ( r ) , where G ( r ) = (cid:12)(cid:12)(cid:12)(cid:12) p − δp (cid:12)(cid:12)(cid:12)(cid:12) p − (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) p − δp (cid:12)(cid:12)(cid:12)(cid:12) r − ( δ − pp − (cid:12)(cid:12)(cid:12) − r p − δp − (cid:12)(cid:12)(cid:12) − p + ( n − δ ) r − δ (cid:12)(cid:12)(cid:12) − r p − δp − (cid:12)(cid:12)(cid:12) − p (cid:21) . Since • for 0 < δ < pG ( r ) = g ( r ) = (cid:18) p − δp (cid:19) p − (cid:20) p − δp r ( − δ +1) pp − (cid:16) − r p − δp − (cid:17) − p + ( n − δ ) r − δ (cid:16) − r p − δp − (cid:17) − p (cid:21) . (6.39) • for 1 < p < δ < nG ( r ) = g ( r ) = (cid:18) δ − pp (cid:19) p − (cid:20) δ − pp r − p (cid:16) − r δ − pp − (cid:17) − p + ( n − δ ) r − p (cid:16) − r δ − pp − (cid:17) − p (cid:21) . It is clear that lim r → G ( r ) = lim r → G ( r ) = ∞ and the positive function G ( r ) has a positiveminimum in (0 , G ( r ) for r ∈ (0 , G ( r ) and • for 0 < δ < p we denote r p − δp − = z, z ∈ (0 ,
1) we get ∂G ( r ) ∂r = ∂g ( r ) ∂r = (cid:18) p − δp (cid:19) p − (cid:20) − p − δp − δ − r − δpp − (cid:16) − r p − δp − (cid:17) − p + ( p − δ ) p − r (1 − δ )( p +1) p − (cid:16) − r p − δp − (cid:17) − p − − δ ( n − δ ) r − δ − (cid:16) − r p − δp − (cid:17) − p + ( p − δ )( n − δ ) r − δ − r p − δp − (cid:16) − r p − δp − (cid:17) − p (cid:21) = (cid:18) p − δp (cid:19) p − r − δ − (cid:16) − r p − δp − (cid:17) − p − (cid:20) − p − δp − δ − z (1 − z )+ ( p − δ ) p − z − δ ( n − δ )(1 − z ) + ( p − δ )( n − δ ) z (1 − z ) (cid:21) = (cid:18) p − δp (cid:19) p − p − z − ( δ +1)( p − p − δ (1 − z ) − − p ( Az + Bz + C ) . for 1 < p < δ < n we denote r δ − pp − = y, y ∈ (0 ,
1) and we get ∂G ( r ) ∂r = ∂g ( r ) ∂r = (cid:18) δ − pp (cid:19) p − (cid:20) − ( δ − p ) r − p − (cid:16) − r δ − pp − (cid:17) − p + ( δ − p ) p − r − p − δ − pp − (cid:16) − r δ − pp − (cid:17) − p − − p ( n − δ ) r − p − (cid:16) − r δ − pp − (cid:17) − p + ( δ − p )( n − δ ) r − p − δ − pp − (cid:16) − r δ − pp − (cid:17) − p (cid:21) = 1 p − (cid:18) δ − pp (cid:19) p − r − p − (cid:16) − r δ − pp − (cid:17) − p − (cid:2) − ( p − δ − p )(1 − z ) + ( δ − p ) z − p ( p − n − δ )(1 − z ) + ( p − δ − p )( n − δ ) z (1 − z ) (cid:3) = (cid:18) δ − pp (cid:19) p − p − y − ( p +1)( p − p − δ (1 − y ) − p − ( Cy + By + A ) , where A , B , C are defined in (6.30).Suppose that A = 0, i.e., δ = p n − p − G ( r ),we have to solve the quadratic equations in (6.31).The discriminant of the equations in (6.31) for ∂G ( r ) ∂r = 0 is D given in (6.33).Since D > A = 0 the equation P ( z ) = Az + Bz + C = 0 (6.40)has two real roots z ± = − B ± | p − δ |√ D A = 2 C − B ∓ | p − δ |√ D , for 0 < δ < p. Analogously, from
C >
D > P ( y ) = Cy + By + A = 0has two real roots y ± = − B ± | p − δ |√ D C = 2 A − B ∓ | p − δ |√ D , for 1 < p < δ < n. Later on we will use only the roots of P ( z ) = 0 and P ( y ) = 0 which are in theinterval (0 , Proposition 6.1.
Let n ≥ , p > , δ ∈ (0 , n ) , δ = p , then the following statements holdi) If p > n , then A ( p, n, δ ) > if and only if p ( n − p − < δ < n so that z − < < z + and inf r ∈ (0 , G ( r ) = g (cid:18) z p − p − δ + (cid:19) ; i2) A ( p, n, δ ) < if and only if < δ < p ( n − p − so that < z + < z − and inf r ∈ (0 , G ( r ) = g (cid:18) z p − p − δ + (cid:19) ; i3) A ( p, n, δ ) = 0 if and only if < δ = p ( n − p − so that (6.40) has an uniquepositive root z + (cid:18) p, n, p ( n − p − (cid:19) = − C ( p, n, p ( n − p − ) B ( p, n, p ( n − p − ) and inf r ∈ (0 , G ( r ) = g z ( p − p ( p − n ) + (cid:18) p, n, p ( n − p − (cid:19)! ; ii) If < p < n then A ( p, n, δ ) < for δ ∈ (0 , n ) andii1) for < δ < p we have < z + < z − and inf r ∈ (0 , G ( r ) = g (cid:18) z p − p − δ + (cid:19) ; ii1) for < p < δ < n we have < y + < y − and inf r ∈ (0 , G ( r ) = g (cid:18) y p − δ − p + (cid:19) ; Proof. i) Since for n > p we have n > p ( n − p − < p < n the inequality p ( n − p − > n holds, the statements for the sign of A ( p, n, δ ) follow immediatelyafter (6.30)i1) From A > , C < B − AC > B and z − = − B − √ D A < − B − | B | A ≤ z + ≥ − B + | B | A = 0. Thus the minimum of G ( r ) in the interval(0 ,
1) is attained at the point z p − p − δ + ;i2) From P (0) = C <
0, lim z →∞ P ( z ) = −∞ we get that z + > , z − > z − = − B + √ D A > − B − √ D A = z + . Since P ( z ) < z ∈ (0 , z + ) and P ( z ) > z ∈ ( z + , z − ), it follows that G ( r ) has a minimum at the point z p − p − δ + ; 823) The proof is trivial.ii) ii1) The proof is identical with the proof of i2);ii2) Since P (0) = A <
0, lim z →∞ P ( z ) = −∞ it follows that y + > , y − > y − = − B + √ D C > − B − √ D C = y + . From the sign of P ( z ) we get that g ( r )attains its minimum at the point y p − δ − p + .(ii) For the case δ = p < n , from (2.62) we get Z B R |∇ u | P dx ≥ Z B R g ( | x | ) | u | p dx, where g ( | x | ) = (cid:18) p − p (cid:19) p | x | − p (cid:12)(cid:12)(cid:12)(cid:12) ln R | x | (cid:12)(cid:12)(cid:12)(cid:12) − p + (cid:18) p − p (cid:19) p − ( n − p ) | x | − p (cid:12)(cid:12)(cid:12)(cid:12) ln R | x | (cid:12)(cid:12)(cid:12)(cid:12) − p . For | x | = Rr, r ∈ (0 ,
1) we obtain g ( | x | ) = g ( Rr ) = R − p G ( r ) , where G ( r ) = (cid:18) p − p (cid:19) p − (cid:26) p − p r − p ( − ln r ) − p + ( n − p ) r − p ( − ln r ) − p (cid:27) . Tedious calculations give us ∂G∂r = (cid:18) p − p (cid:19) p − p z − p − e − ( p +1) z [ A z + B z + C ] , for z = − ln r > . Critical points of ∂G∂r are the solutions of the quadratic equation (6.32).Since D ( p, n ) > A ( p, n ) = 0 equation (6.32) has two real roots z ± = − B ± √ D A , z < z − . From P (0) = p ( p − > z →∞ P ( z ) = −∞ it follows that z < < z − .Thus G ( r ) attains its minimum in (0 ,
1) at the point r = e − z − , where from (6.34) z − = 2 C hp D − B i − . Theorem 6.4 is proved. λ p,n ( B R ) In this section we will compare analytical estimates from below of λ p,n ( B R ) defined inTheorem 6.4 Λ (3) p,n ( B R ) with Λ (1) p,n ( B R ), defined in (6.4) with Λ (2) p,n ( B R ), defined in (6.6) andwith Λ ( H ) p,n ( B R ), Λ ( L ) p,n ( B R ), defined in (6.10) and (6.8). We compare only those estimatesof λ p,n ( B R ) that are given with analytical formulas for every p > n ≥
2. The estimatein (6.21), Sect. 6.2 is valid only for p > n ≥ (3 , p,n ( B R ). 83 omparison of Λ ( H ) p,n ( B R ) and Λ ( L ) p,n ( B R ) with Λ (1) p,n ( B R ) , Λ (2) p,n ( B R ) and Λ (3) p,n ( B R )Let us compare Λ ( H ) p,n ( B R ) = (cid:12)(cid:12)(cid:12)(cid:12) n − ppR (cid:12)(cid:12)(cid:12)(cid:12) p for n ≥ p > n = p with other lower bounds for λ p,n ( B R ). • For p > n ≥ ( H ) p,n ( B R ) = (cid:18) p − npR (cid:19) p = 1 R p (cid:18) − np (cid:19) p < R p < nR p = Λ (3 , p,n ( B R ); • For 1 < p < n the inequalityΛ ( H ) p,n ( B R ) = (cid:18) n − ppR (cid:19) p < (cid:18) npR (cid:19) p = Λ (1) p,n ( B R )holds. • As for the Lindqvist’s constant Λ ( L ) p,n ( B R ) = pR p for p > n ≥ ( L ) p,n ( B R ) = pR p < npR p = Λ (2 , p,n ( B R ) . Comparison of Λ (3) p,n ( B R ) with Λ (1) p,n ( B R )We will use the estimate (6.36), which coincides with the estimate (6.13). Proposition 6.2.
For every n ≥ there exists p n , < p n < such that Λ (1) p,n < Λ (3 , p,n ≤ Λ (3) p,n , for p n < p. Proof.
We define the function f n ( p ) = 1 n − p [( n −
1) ln( n − − ( p −
1) ln( p − − ln n. The inequality Λ (3 , p,n ( B R ) > Λ (1) p,n ( B R ) holds if and only if f n ( p ) >
0. We will show that forevery fixed n ≥ f n ( p ) is astrictly increasing one for p > p → f n ( p ) < f n (2) > n ≥
2, lim n → f n (2) = 1 − ln 2 >
0. Thus, there exists p n ∈ (1 ,
2) such that f n ( p ) < < p < p n , f n ( p n ) = ( n −
1) ln( n − − ( p n −
1) ln( p n − − ( n − p n ) ln n = 0 , (6.41)and f n ( p ) > p n < p .For the first derivative of f n ( p ) we have f ′ n ( p ) = 1( n − p ) [( n −
1) ln( n − − ( n −
1) + ( p − − ( n −
1) ln( p − g n ( p )( n − p ) . g ′ n ( p ) = p − np − g ′′ n ( p ) = n − p − > g n ( p ) has a minimum at the point p = n and g n ( n ) = 0. Using L’Hospital rule we obtain lim p → n f ′ n ( p ) = 12( n − > f ′ n ( p ) > p >
1. Moreover, lim p → f n ( p ) = ln( n − − ln n <
0, and f n (2) = 1 n − n −
1) ln( n − − ( n −
2) ln n ] > . (6.42)The inequality (6.42) holds because for the function z ( n ) = ( n −
1) ln( n − − ( n −
2) ln n we have z ′ = 2 n + ln( n − − ln n , z ′′ = 2 − nn ( n − ≤
0, i.e., z ′ is a decreasing function, z ′ ( n ) > lim n →∞ z ′ ( n ) >
0. Hence z ( n ) is a strictly increasing function and z ( n ) > z (2) =0. Comparison of Λ (3) p,n ( B R ) with Λ (2) p,n ( B R ) Proposition 6.3.
For integer n ≥ and p ≥ p n = 278 (cid:18) n n − (cid:19) the estimate Λ (2) p,n ( B R ) < Λ (3) p,n ( B R ) , (6.43) holds.Proof. From (6.38) and (6.39) for δ < p , δ ∈ (0 , n ) it follows thatΛ (3) p,n ( B R ) = R − p sup δ ∈ (0 ,n ) inf r ∈ (0 , g ( r ) ≥ R − p sup δ ∈ (0 ,n ) inf r ∈ (0 , H ( r ) , where H ( R ) = ( n − δ ) (cid:18) p − δp (cid:19) p − r − δ (cid:16) − r p − δp − (cid:17) − p . The positive function h ( r ) = 1 p − δ r δp − (cid:16) − r p − δp − (cid:17) attains its maximum for 0 ≤ r ≤ r = (cid:18) δp (cid:19) p − p − δ <
1, because h ′ ( r ) = r δ − p +1 p − ( p − p − δ ) h δ − pr p − δp − i ,h ′ ( r ) = 0 , h (0) = h (1) = 0 . Thus from the equalitiesinf r ∈ (0 , H ( r ) = ( n − δ ) inf r ∈ (0 , ( ph ( r )) − p = ( n − δ ) sup r ∈ (0 , ( ph ( r )) p − = ( n − δ ) ( ph ( r )) p − = ( n − δ ) (cid:16) pδ (cid:17) δ ( p − p − δ
85e get the estimateΛ (3) p,n ( B R ) ≥ Λ (3 , p,n ( B R ) = R − p sup δ ∈ (0 ,n ) ( n − δ ) (cid:16) pδ (cid:17) δ ( p − p − δ . (6.44)Thus from (6.44), the estimate (6.43) holds ifΛ (2) p,n ( B R ) < Λ (3 , p,n ( B R ) , n ≥ , p ≥ p n . (6.45)Note that p n is a decreasing function for n ∈ [2 , ∞ ), so 3 . < p n < p n > n ≥ δ < p the estimate (6.45) is equivalent to the inequalitysup δ ∈ (0 ,n ) ,δ
np for p ≥ p n , p > δ .For δ = 32 and p ≥ p n a simple computation gives ussup δ ∈ (0 ,n ) ,δ
(cid:18) n − (cid:19) (cid:18) p (cid:19) p − p − = 12 (2 n − (cid:18) p (cid:19) (cid:18) p (cid:19) p − ≥
13 (2 n − (cid:18) p (cid:19) p ≥
13 (2 n − (cid:18) p ,n (cid:19) p = 13 (2 n −
3) 32 2 n n − p = np. Comparison of Λ (2 , p,n ( B R ) with Λ (2 , p,n ( B R ) Proposition 6.4.
For every n ≥ the estimates Λ (2 , p,n ( B R ) > Λ (2 , p,n ( B R ) for p ∈ (1 , , (6.46)Λ (2 , p,n ( B R ) < Λ (2 , p,n ( B R ) for p > , (6.47) hold.Proof. The inequality (6.46) is equivalent to h ( p ) = ( p −
1) ln pp − − ln p > p ∈ (1 , h ( p ) < p > h ′ ( p ) = ln pp − − p , h ′′ ( p ) = p − p ( p − , h ′ (2) = ln 2 e < , (6.48)and from the L’Hospital rulelim p →∞ h ′ ( p ) = lim p →∞ p ln pp − − p = lim p →∞ (cid:18) ln pp − − p − (cid:19) = 0 . Thus we have from (6.48) that h ′ ( p ) < p >
2. From h (2) = 0 it follows that h ( p ) < p > p → h ( p ) = 0 and the concavity of h ( p ) for p ∈ (1 ,
2) we get h ( p ) > p ∈ (1 , omparison of Λ (1) p,n ( B R ) with Λ (2) p,n ( B R )According to Proposition 6.4 we will compare Λ (1) p,n ( B R ) with Λ (2 , p,n ( B R ) for p ∈ (1 , (1) p,n ( B R ) with Λ (2 , p,n ( B R ) for p ≥ Proposition 6.5. • If n ∈ [2 , , then Λ (2) p,n ( B R ) > Λ (1) p,n ( B R ) for p > , p = 2 and Λ (2)2 , ( B R ) > Λ (1)2 , ( B R ) . • If n ≥ , then there exist constants p ,n ∈ (1 , and p ,n ≥ such that Λ (2) p,n ( B R ) > Λ (1) p,n ( B R ) for p ∈ (1 , p ,n ) ∪ ( p ,n , ∞ ) , Λ (2) p,n ( B R ) < Λ (1) p,n ( B R ) for p ∈ ( p ,n , p ,n ) . (6.49) Proof.
Case 1: p ∈ (1 , n , inequality Λ (1) p,n ( B R ) > Λ (2 , p,n ( B R ) is equivalent to h ( p ) = ( p −
1) ln n − (2 p −
1) ln p + ( p −
1) ln( p − > , for p ∈ (1 , . Simple computations give us h ′ ( p ) = ln n − p − p + ln( p − h ′′ ( p ) = − p − p − p ( p − h ′′ ( p ) > p ∈ , √ ! , h ′′ ( p ) < p ∈ ( √ , ! . Hence h ′ ( p ) has amaximum at the point √ h ′ √ ! = ln n − √ − √ √ −
12= ln n − √ − ! − ln √ ! , i.e., h ′ √ ! < n < √ ! e √ − ≈ . h ′ √ ! > n > . . Since lim p → h ′ ( p ) = −∞ , h ′ (2) = 12 ln n e < n ∈ [2 ,
6] and h ′ (2) > n ≥
7, itfollows that h ′ ( p ) < , for n ∈ [2 ,
6] and p ∈ (1 , . For n ≥ q n ∈ , √ ! such that h ′ ( p ) < , for p ∈ (1 , q n ) ,h ′ ( p ) > , for p ∈ ( q n , . p → h ( p ) = 0, h (2) = ln n h (2) < n ∈ [2 , h (2) = 0 for n = 8, h (2) > n > h ( p ) < p ∈ (1 , n ∈ [2 , h (2) = ln n > n ≥
9, there exists p ,n ∈ (1 ,
2) such that h ( p ,n ) = ( p ,n −
1) ln n − (2 p ,n −
1) ln p ,n + ( p ,n −
1) ln( p ,n −
1) = 0 , (6.50)and h ( p ) < p ∈ (1 , p ,n ) and h ( p ) > p ∈ ( p ,n , p ∈ (1 ,
2) is proved.Case 2: p ≥ p ≥ n ≥ h ( p ) = ( p −
1) ln n − ( p + 1) ln p > , for p ≥ . A simple computation gives us h ′ ( p ) = ln n − ln p − p + 1 p , h ′′ ( p ) = 1 − pp < p ≥ h (2) = ln n < n ∈ [2 , h (2) = 0, for n = 8.Since h ′ (2) = ln n e / ≤ ln 82 e / ≈ − . < n ∈ [2 ,
8] it follows that h ( p ) < p > n ∈ [2 , n ≥ h (2) = ln n >
0, lim p →∞ h ( p ) = −∞ and consequently there exists aconstant p ,n > h ( p ,n ) = ( p n −
1) ln n − ( p n + 1) ln p ,n = 0 , such that h ( p ) > p ∈ [2 , p ,n ), h ( p ) < p > p ,n . For example, p ,n = 3 for n = 9.Finally, we summarize the analytical results in Propositions 6.2–6.5.Suppose n ≥
9, then from Proposition 6.2 we get Λ (3) p,n ( B R ) > Λ (1) p,n ( B R ) for p > p n ,where p n ∈ (1 ,
2) is a solution of equation (6.41).Analogously, from Proposition 6.5 we obtain the estimates Λ (1) p,n ( B R ) > Λ (2) p,n ( B R ) for p > p n and Λ (1) p,n ( B R ) < Λ (2) p,n ( B R ) for p ∈ (1 , p n ), where p n ∈ (1 ,
2) is a solution of theequation (6.50). Thus for n ≥ p ,n < p ,n thenΛ (2) p,n ( B R ) > max n Λ (1) p,n ( B R ) , Λ (3) p,n ( B R ) o , for p ∈ (1 , p ,n ) , Λ (1) p,n ( B R ) > Λ (2) p,n ( B R ) , for p ∈ ( p ,n , p ,n ) , Λ (3) p,n ( B R ) > max n Λ (1) p,n ( B R ) , Λ (2) p,n ( B R ) o , for p > p ,n . (ii) If p ,n < p ,n then there exists p ,n ∈ [ p ,n , p ,n ) ⊂ (1 ,
2) such thatΛ (2) p,n ( B R ) > max n Λ (1) p,n ( B R ) , Λ (3) p,n ( B R ) o , for p ∈ (1 , p ,n ) , Λ (3) p,n ( B R ) > max n Λ (1) p,n ( B R ) , Λ (2) p,n ( B R ) o , for p > p ,n . < p < p n , n ∈ [2 ,
8] the comparison is by means of numerical calculations.
Remark 6.1.
For sufficiently large values of n we get p ,n > p ,n . Indeed, after the limit n → ∞ in (6.50) it follows that lim n →∞ p n = 1. From the definitions of p ,n and p ,n wehave that p = p ,n satisfying the equation( n − p ) f n ( p ) = ( n −
1) ln( n − − ( p −
1) ln( p − − ( n − p ) ln n = 0 , while y = p ,n satisfies the equation h ( p ) = ( y −
1) ln n − (2 y −
1) ln y + ( y −
1) ln( y −
1) = 0 . Hence for y we obtain( n − p ) f n ( y ) − h ( y ) = ( n −
1) ln n − n + (2 y + 1) ln y − y −
1) ln( y − → n →∞ − , because( n −
1) ln n − n = (cid:18) nn − (cid:19) ln (cid:18) n − n (cid:19) n → n →∞ ln e − = − , and lim y → ( y −
1) ln( y −
1) = 0 . From the inequality f n ( y ) = f n ( p ,n ) < n sufficiently large it follows that p ,n < p ,n for n ≫ Numerical comparison of Λ (3) p,n ( B R ) and Λ (2) p,n ( B R )Using the formulas (6.35) for Λ (3) p,n ( B R ) and (6.6) for Λ (2) p,n ( B R ) we listed below in Table1 for R = 1 and fixed n ∈ [2 ,
9] the intervals of p where Λ (3) p,n ( B ) geq Λ (2) p,n ( B ) and whereΛ (2) p,n ( B ) ≥ Λ (3) p,n ( B ). Numerical calculations are made by Mathematica 6. For example,for n = 3 and p > .
25 we have Λ (3) p,n > Λ (2) p,n , while for p < .
25 we have Λ (3) p,n < Λ (2) p,n .Table 1: Numerical comparison of Λ (3) p,n and Λ (2) p,n for R = 1, n = 2 , . . . , p > . n p ≈ . p ≈ . p ≈ . p ≈ . p ≈ . p ≈ . p ≈ . p ≈ .
682 Λ (2) p,n ← | → Λ (3) p,n (2) p,n ← | → Λ (3) p,n (2) p,n ← | → Λ (3) p,n (2) p,n ← | → Λ (3) p,n (2) p,n ← | → Λ (3) p,n (2) p,n ← | → Λ (3) p,n (2) p,n ← | → Λ (3) p,n (2) p,n ← | → Λ (3) p,n Comparison of Λ (3 , p,n ( B R ) with numerical values As is mention in Sect. 6.1 iterative numerical method for evaluating the first eigenvaluewas developed in Biezuner et al. [22, 23] where the approximate values, denoted here asΛ ( num ) p,n ( B ) of the first eigenvalue Λ p,n ( B ) are given for p ∈ (1 ,
4] and n = 2 , , λ p,n ( B )and the estimates from below Λ (3 , p,n ( B ) is about 2.5 times more. Nevertheless the pre-sented method for estimates of λ p,n ( B ) from below using Hardy inequality with doublesingular weights gives analytical estimates for every p > n ≥ (3 , p,n and numerical values Λ ( num ) p,n in Table 1, Biezuneret al. [23] p n = 2 n = 3 n = 4Λ (3 , p,n Λ ( num ) p,n Λ (3 , p,n Λ ( num ) p,n Λ (3 , p,n Λ ( num ) p,n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
000 9 . . . . . . . .
405 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement
This paper has been accomplished with the financial support byGrant No BG05M2OP001-1.001-0003, financed by the Science and Education for SmartGrowth Operational Program (2014-2020) in Bulgaria and co-financed by the EuropeanUnion through the European Structural and Investment Funds and also by the Na-tional Scientific Program ”Information and Communication Technologies for a Single Dig-ital Market in Science, Education and Security (ICTinSES)”, under contract No DO1-205/23.11.2018.
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