On the Dirichlet Problem Generated by the Maz'ya--Sobolev Inequality
aa r X i v : . [ m a t h . A P ] J a n ON THE DIRICHLET PROBLEMGENERATED BY THE MAZ’YA–SOBOLEV INEQUALITY
A.I. Nazarov ∗ ,Saint-Petersburg State University, e-mail: [email protected] In what follows, x = ( y ; z ) = ( y , y ′ ; z ) stands for a point in R n = R m × R n − m , n ≥ ≤ m ≤ n −
1. Denote by P the subspace { x ∈ R n : y = 0 } ; correspondingly, P ⊥ = { x ∈ R n : z = 0 } .Let Ω be a domain in R n . By C ∞ (Ω) we denote the set of smooth functions with compactsupport in Ω. For 1 ≤ p < ∞ we denote by ˙ W p (Ω) the closure of C ∞ (Ω) with respect to thenorm k∇ v k p, Ω . Obviously, for bounded domains ˙ W p (Ω) = o W p (Ω).By definition, for 0 ≤ σ ≤ min { , np } we put p ∗ σ = npn − σp . Proposition
The following inequality k| y | σ − v k p ∗ σ , Ω ≤ N ( p, σ, Ω) · k∇ v k p, Ω . (1) holds true for any v ∈ ˙ W p (Ω) provided a ) Ω is any domain in R n for n ( p − m ) p ( n − m ) < σ ≤ (the region I on Fig. 1) ; b ) Ω ⊂ R n \ P for p > m, σ ≤ min { n ( p − m ) p ( n − m ) ; np } , σ = 1 ( II on Fig. 1) ; c ) Ω ⊂ R n \ ( ℓ × R n − m ) for p = m, σ = 0 (black point on Fig. 1) (2) (here ℓ is a ray in R m beginning at the origin). Proof . The case a) is well known; see, e.g., [13, Sec.2.1.6]. Note that for σ = 1 we haveclassical Sobolev inequality.Consider the cases b) and c). Note that it is sufficient to prove (1) for Ω = R n \ P (respectively, Ω = R n \ ( ℓ × R n − m )).For σ = 0 one should take conventional Hardy inequality in R m \ { } (respectively, in R m \ ℓ ; see, e.g., [16, Sec.2]) and integrate it with respect to z .For m < p < n the inequality (1) can be obtained from the cases σ = 0 and σ = 1 by theH¨older inequality. For p > n we also obtain (1) by the H¨older inequality from the extremecases σ = 0 and σ = np ; the last one corresponds to the Morrey inequality, see [13, Sec.1.4.5]. ∗ Supported by grant NSh.4210.2010.1. ✻ m n p σ III ✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄ ❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊ ❜ r
Figure 1: To the Proposition 1.1Finally, we deal with the case p = n , 0 < σ <
1. Consider the domain Ω = Ω × Ω ,where Ω = B \ B ⊂ R m is a spherical layer while Ω =]0 , n − m ⊂ R n − m is a cube. Let uswrite down the embedding theorem W n (Ω) ֒ → L q (Ω), with q = n − σ . Since the function | y | n is bounded and separated from zero in Ω, this theorem can be rewritten as follows: Z Ω | v | q | y | n dx ! n/q ≤ C ( q, m, n ) Z Ω (cid:18) |∇ v | n + | v | n | y | n (cid:19) dx. Note that all the terms in this inequality are invariant under translations in z and underdilations in x . Therefore, the same inequality is valid for Ω k k = 2 k (cid:0) Ω × (Ω + k ) (cid:1) , with k ∈ Z , k ∈ Z n − m . Summing these inequalities we obtain, subject to q > n , Z R n | v | q | y | n dx ! n/q ≤ X k, k Z Ω k k | v | q | y | n dx ! n/q ≤ C ( q, m, n ) Z R n (cid:18) |∇ v | n + | v | n | y | n (cid:19) dx. The last term is already estimated, and we arrive at (1). (cid:3)
Remark The assumption on Ω in the case c) can be considerably weakened. However, itis sharp for Ω being a wedge. We call (1) the Maz’ya–Sobolev inequality .We are interested in the attainability of the sharp constant in (1), i.e. in the attainabilityof the norm of corresponding embedding operator. If Ω is unbounded, or Ω ∩ P 6 = ∅ , then2his operator is, in general, noncompact; for p < n and σ = 1 this is the case for any Ω.Therefore, the problem of attainability is nontrivial.It is well known that the sharp constant in the Sobolev inequality ( p < n and σ = 1)does not depend on Ω and is not attained for any Ω provided the complement of Ω is notnegligible, i.e. ˙ W p (Ω) = ˙ W p ( R n ). We claim that the same is true for p < n and 0 < σ < ∩ P 6 = ∅ . Indeed, since the inequality (1) is dilation invariant, the sharp constantin this case cannot depend on Ω and equals N ( p, σ, R n ). Further, if the extremal functionin (1) exists, by standard argument (see, for example, the end of the proof of Theorem 2.2)it is (after a suitable normalization) a positive generalized solution of the Dirichlet problem − ∆ p u = u p ∗ σ − | y | (1 − σ ) p ∗ σ in Ω; u (cid:12)(cid:12) ∂ Ω = 0 (3)(here ∆ p u = div( |∇ u | p − ∇ u ) is p -Laplacian).Extending u by zero to R n , we obtain an extremal for (1) in the whole space. Therefore,this extension solves the equation (3) in R n , and thus it is positive in R n , a contradiction.By the way, it is worth to note that for p = n the exponent in the denominator of (3)does not depend on σ and equals n .The case Ω ∩ P = ∅ , ∂ Ω ∩ P 6 = ∅ is considerably more complicated. In the recent paper[8] the attainability of the sharp constant in (1) was proved for p = 2, 0 < σ <
1, underrather restrictive assumptions on (a smooth bounded domain) Ω. Namely, it is supposed in[8, Theorem 1.1] that all the main curvatures at any point x ∈ ∂ Ω ∩ P are nonpositive, andthe mean curvature at any such point does not vanish.Our paper consists of two parts. First, we analyze the attainability of the sharp constantin (1) for Ω being a wedge K = K × R n − m (here K is an open cone in R m ) or a “perturbed”wedge. Here we consider all 1 < p < ∞ and 0 ≤ σ < min { , np } . Naturally, we suppose thatΩ satisfies (2).In the second part we prove the attainability of the sharp constant in (1) in a boundeddomain for p = 2 and 0 < σ < ∂ Ω, seeSection 3 below. Unfortunately, we cannot transfer this result to the case of arbitrary p because we do not have in hands good estimates of solutions to the model problem (3) in ahalf-space.Let us discuss briefly the cases m = 1 and m = n . For m = 1 our problem of interestdegenerates in a sence . Indeed, the only admissible wedge in this case is a half-space R n + = { x ∈ R n : y > } . Theorems 2.1 and 2.2 in this case remain valid with the sameproof while Theorems 2.3 and 2.4 are irrelevant. As for other domains, if Ω ⊂ R n + , and ∂ Ω ∈ C touches P , then in the neighborhood of a touching point x Ω in the large scalelooks like a half-space. Since (1) is dilation invariant, we obtain N ( p, σ, Ω) ≤ N ( p, σ, R n + ).The reverse inequality is trivial. As in the case Ω ∩ P 6 = ∅ , this implies non-attainabilityof the sharp constant in (1) for any Ω provided the complement of Ω is not negligible in R n + . For p = 2 and bounded domain this fact was proved in [8]. Attainability of the sharpconstant for m = 1, p = 2 in some unbounded domains without touching of P was discussedin [23] . Note that Proposition 1.1 holds true for m = 1 with the only exception: the case c) should be attachedto the case b). The proof runs without changes. Example 1 after Lemma 2.7 in [23] is not completely correct; it should be ϕ > ϕ ≥
3n the another hand, the problem for m = n , corresponding to the Hardy–Sobolevinequality, was investigated in a number of papers. The existence of the extremal functionin a cone was proved in [16] (in the case p = 2, n ≥ p = 2, σ = 0was dealt with in [19]). For Ω being a compact Riemannian manifold with boundary, theconditions of attainability of the sharp constants in (1) and in some similar inequalities wereconsidered in [2]. The case of bounded domains with 0 ∈ ∂ Ω was treated in [3] for p = 2, n ≥
2; similar results under more restrictive assumptions on ∂ Ω were obtained earlier inrather involved papers [6] for n ≥ n = 3. See also the survey [17], where thehistory of related problems and extensive bibliography was given.The paper is organized as follows. In Section 2 we collect the results on existence andqualitative properties of extremal functions in (1) in wedges and in wedges with compactperturbation bounded away from P .In Section 3 we formulate the assumptions on the behavior of ∂ Ω in a neighborhood ofthe origin and prove existence theorems for bounded domains. The technical estimates usedin this proof are given in Sections 4–6.Let us introduce the following notation. S n − r is the sphere in R n with radius r centeredat the origin; ω n − = π n/ Γ( n ) is the area of S n − .We write o ε (1) to show the quantity tending to zero, as ε →
0, with other parametersassumed to be fixed. All the other o (1) have the same meaning but are uniform with respectto ε .We recall that a function f :]0 , δ [ → R is regularly varying (RVF) of order α at theorigin, if it has a constant sign, and for any t > ε → f ( εt ) f ( ε ) = t α . For basic properties of RVFs see [22].We use letter C to denote various positive constants. To indicate that C depends onsome parameters, we write C ( . . . ). Our first statement provides the sharp constants in the Maz’ya inequality in wedges.
Theorem
Let ≤ m ≤ n − , < p < ∞ , σ = 0 . Let K be a cone in R m . If p ≥ m wesuppose that K = R m , and for p = m , in addition, K = R m \ { } . Put Ω = K = K × R n − m and G = K ∩ S m − . Then the sharp constant in (1) is not attained and equals (Λ ( p ) ( G )) − p ,where Λ ( p ) ( G ) = min v ∈ o W p ( G ) \{ } Z G (cid:16)(cid:16) m − pp (cid:17) v + |∇ ′ v | (cid:17) p dS Z G | v | p dS (4) (here ∇ ′ stands for the tangential gradient on S m − ⊂ R m ). roof . First, the minimum in (4) is attained due to the compactness of embedding o W p ( G ) ֒ → L p ( G ). Denote by b V the minimizer of (4) normalized in L p ( G ). By standardargument, b V is positive in G .Let us define U ( y, z ) = U ( y ) = | y | − mp · b V (cid:0) y | y | (cid:1) . It is shown in [17, Theorem 18] that U isa positive weak solution of the equation − ∆ ( y ) p U = Λ ( p ) ( G ) U p − | y | p in K, and thus, − ∆ p U = Λ ( p ) ( G ) U p − | y | p in K . (5)The relation Λ ( p ) ( G ) ≤ N − p ( p, , Ω) follows now from [20, Theorem 2.3]. For the reader’sconvenience we reproduce the proof based on the so-called generalized Picone identity.For any u ∈ C ∞ (Ω) we set h = | u | p U p − ∈ C (Ω). Then (5) impliesΛ ( p ) ( G ) Z Ω | u | p | y | p dx = Λ ( p ) ( G ) Z Ω U p − | y | p h dx = Z Ω |∇ U | p − ∇ U · ∇ h dx == Z Ω (cid:16) p |∇ U | p − ∇ U · ∇ u | u | p − uU p − − ( p − |∇ U | p | u | p U p (cid:17) dx ∗ ≤ ∗ ≤ Z Ω (cid:16) p |∇ u | · |∇ U | p − | u | p − U p − − ( p − |∇ U | p | u | p U p (cid:17) dx ≤ Z Ω |∇ u | p dx. (6)Here ( ∗ ) is the Cauchy inequality while the last inequality follows from r p − prt p − + ( p − t p ≥ , r, t > . (7)By approximation, (6) holds true for u ∈ ˙ W p (Ω).To prove Λ ( p ) ( G ) = N − p ( p, , Ω) we consider the sequence u δ ( y, z ) = U δ ( y ) Z δ ( z ), where U δ ( y ) = | y | − mp + δ · b V (cid:0) y | y | (cid:1) , | y | ≤ R,R − mp + δ (cid:0) − | y | R (cid:1) · b V (cid:0) y | y | (cid:1) , R ≤ | y | ≤ R, , | y | ≥ R ; Z δ ( Z ) = , | z | ≤ R, − | z | R , R ≤ | z | ≤ R, , | z | ≥ R. Clearly, u δ ∈ ˙ W p (Ω). Direct computation shows Z Ω |∇ u δ | p dx = Z Ω | u δ | p r p dx · (cid:0) Λ ( p ) ( G ) + O ( δ ) (cid:1) , and the statement follows.Finally, the equality sign in ( ∗ ) means ∇ u k ∇ U while the equality in (7) means r = t .These two facts imply ∇ uu = ∇ UU = ⇒ u = cU on the set { u = 0 } and, therefore, in the whole Ω. Since U / ∈ ˙ W p (Ω), the equality in (6) isimpossible. (cid:3) Next, we consider the Maz’ya–Sobolev inequality in wedges.5 heorem
Let ≤ m ≤ n − , < p < ∞ , < σ < min { , np } . Let K be a cone in R m .If p > m and σ ≤ n ( p − m ) p ( n − m ) we suppose in addition that K = R m . Put Ω = K = K × R n − m .Then the sharp constant in (1) is attained, i.e. there exists a function V ∈ ˙ W p (Ω) , V > in Ω , such that the inequality (1) becomes equality. Proof . It is evident that the sharp constant in (1) satisfies the relation N − ( p, σ, Ω) = inf v ∈ ˙ W p (Ω) \{ } J ( v ) ≡ inf v ∈ ˙ W p (Ω) \{ } k∇ v k p, Ω k| y | σ − v k p ∗ σ , Ω . (8)Let { v k } be a minimizing sequence for the functional J . Without loss of generality wecan assume k| y | σ − v k k p ∗ σ , Ω = 1 and v k ⇁ v in ˙ W p (Ω). By the concentration-compactnessprinciple of Lions ([11]; see also [5, Ch.1]) we have || y | σ − v k | p ∗ σ ⇁ || y | σ − v | p ∗ σ + X j ∈M α j δ ( x − x j ) , |∇ v k | p ⇁ M ≥ |∇ v | p + N − p ( p, σ, Ω) X j ∈M α p/p ∗ σ j δ ( x − x j ) , where the convergence is understood in the sense of measures on the one-point compactifi-cation Ω ∪ {∞} , a set M is at most countable and α j >
0. Moreover, since the embedding˙ W p (Ω) ֒ → L p ∗ σ (Ω) is locally compact, we conclude that x j ∈ P ∪ {∞} .Since { v k } is a minimizing sequence, by verbatim repetition of arguments from Theorem2.2 [12] we obtain the alternative — either v k → v in ˙ W p (Ω) and M = ∅ (in this case v is aminimizer of J ), or v = 0, M is a singleton and α = 1.Let us remark here that, by the dilation invariance of the functional J , we can ensurethe additional relation R Ω ∩ B || y | σ − v k | p ∗ σ dx = , which takes away the second variant.It remains to note that the function V = | v | also provides the minimum in the problem(8). Thus, after multiplying by a suitable constant, V becomes a nonnegative generalizedsolution of the Dirichlet problem to the Euler–Lagrange equation (3) and thus, it is super- p -harmonic in Ω. By the Harnack inequality for p -harmonic functions (see, e.g., [24]), it ispositive in Ω. (cid:3) Now we present some symmetry properties of the extremal function.
Theorem
Let the assumptions of Theorem 2.2 be fulfilled. Then the functon V providingthe sharp constant in (1) has the following properties:1. V is radially symmetric with respect to z , i.e. V = V ( y ; | z | ) ;2. If K is a circular cone, then V is radially symmetric with respect to y ′ and z , i.e. V = V ( y , | y ′ | ; | z | ) ;3. If K = R m and σ > n ( p − m ) p ( n − m ) , then V is radially symmetric with respect to y and z , i.e. V = V ( | y | ; | z | ) ;4. Let K = R m \ { } . There exists b p ∈ ] m, n [ , and for p > b p the function b σ ( m, n, p ) is defined, such that b σ < min { , np } and for σ > b σ the function V is not radiallysymmetric w.r.t. y . roof . 1. This statement follows from the properties of the Schwarz symmetrizationwith respect to z -variables (or from the properties of the Steiner symmetrization with respectto z for m = n − z . Further, by the Eulerequation (3) all critical points of an extremal radially symmetric w.r.t. z have to be locatedat P . In this case the numerator in (8) strictly decreases under symmetrization (see [1]),and therefore no function asymmetric w.r.t. z can provide the minimum in (8).2. In addition to the Part 1, in this case we can apply spherical symmetrization alongthe spheres S m − r , which does not enlarge the numerator, see, e.g., [21, App.C], and retainsthe denominator.3. Here we can apply the Schwarz symmetrization with respect to y -variables which doesnot enlarge the numerator, and does not reduce the denominator, see, e.g., [10, Ch.3].4. In this case the Schwarz symmetrization in y s does not work, and we show that theminimizer in general does not inherit the symmetry of extremal problem.Let u ( | y | ; | z | ) be a function providing the minimum to the functional J over the set offunctions in ˙ W p (Ω), radially symmetric w.r.t. y and z . Without loss of generality, we assumethat k| y | σ − u k p ∗ σ , Ω = 1. By the principle of symmetric criticality, see [18], dJ σ ( u ; h ) = 0 forany variation h ∈ ˙ W p (Ω).Similarly to [15, Theorem 1.3], the second differential of J at the point u can be writtenas follows: J p − ( u ) · d J ( u ; h ) = Z Ω |∇ u | p − (cid:0) ( p − h∇ u, ∇ h i + |∇ u | |∇ h | (cid:1) dx −− J p ( u ) · h ( p − p ∗ σ ) · (cid:16) Z Ω | u | p ∗ σ − uh | y | (1 − σ ) p ∗ σ dx (cid:17) + ( p ∗ σ − · Z Ω | u | p ∗ σ − h | y | (1 − σ ) p ∗ σ dx i . (9)Now we set h ( y ; z ) = u ( | y | ; | z | ) · y | y | . By symmetry of u , R Ω | u | p ∗ σ − uh | y | (1 − σ ) p ∗ σ dx = 0. Substitutinginto (9), we obtain J p − ( u ) · d J ( u ; h ) = Z Ω |∇ u | p − u | y | dx − J p ( u ) · p ∗ σ − pm − · Z Ω | u | p ∗ σ f | y | (1 − σ ) p ∗ σ dx. Finally, we estimate the first integral by H¨older and Hardy inequalities and arrive at d J ( u ; h ) ≤ J ( u ) · "(cid:18) pp − m (cid:19) − p σ ( m − n − pσ ) . If p ≥ n then the quantity in square brackets is negative for σ close to np . If p < n is closeto n , this quantity is also negative for σ close to 1. In both cases the statement follows. (cid:3) Corollary . For p > b p and b σ < σ < min { , np } the problem (3) in R n \ P has at least twononequivalent positive solutions. Proof . The first solution is a global minimizer of J (under suitable normalization), thesecond one is a minimizer over the set of functions symmetric w.r.t. y . (cid:3) Further, we consider Ω being a perturbed wedge.7 heorem
Suppose that ≤ m ≤ n − , < p < ∞ and ≤ σ < min { , np } . Let Ω = K = K × R n − m be a wedge satisfying (2), Ω ⋐ R n \ P and Ω ∩ Ω = ∅ . . For Ω = Ω \ Ω is not attained. . Let σ > . Then for Ω = Ω ∪ Ω the sharp constant in (1) is attained provided ˙ W p (Ω) = ˙ W p (Ω ) . . Let σ = 0 . Then, given Ω ′ ⋐ R m \ { } , Ω ′ ∩ K = ∅ , there exists L < ∞ such that if Ω ⊃ Ω ′ × ] − L, L [ , Ω = Ω ∪ Ω and ˙ W p (Ω) = ˙ W p (Ω ) then the sharp constant in (1) isattained. Proof . . For any u ∈ C ∞ (Ω ) there exists a dilation Π such that Π u ∈ C ∞ (Ω). Due tothe dilation invariance of (1) we conclude that N ( p, σ, Ω) = N ( p, σ, Ω ).Thus, if u minimizes the quotient (8) on ˙ W p (Ω) then its zero continuation minimizes (8)on ˙ W p (Ω ). Therefore, it is the nonnegative solution of the problem (3) in Ω . By Harnack’sinequality for p -harmonic functions, it is positive in Ω , a contradiction. . By Theorem 2.2, there exists a function u positive in Ω that minimizes the quotient(8) on ˙ W p (Ω ). If N ( p, σ, Ω) = N ( p, σ, Ω ) then the zero continuation of u minimizes (8) on˙ W p (Ω) that again leads to contradiction. Therefore, N ( p, σ, Ω) > N ( p, σ, Ω ).Now the statement follows by the concentration-compactness principle. Indeed, let { v k } be a minimizing sequence for the functional J . Without loss of generality we can assume k| y | σ − v k k p ∗ σ , Ω = 1 and v k ⇁ v in ˙ W p (Ω). As in Theorem 2.2, if v k v then || y | σ − v k | p ∗ σ ⇁ δ ( x − b x ) , |∇ v k | p ⇁ N − p ( p, σ, Ω) δ ( b x − x ) , and b x ∈ P ∪ {∞} .Since Ω ⋐ R n \ P , similarly to the proof of Corollary 2.1 [12], we can assume that v k (cid:12)(cid:12) Ω ≡
0. This implies N ( p, σ, Ω) ≤ N ( p, σ, Ω ), a contradiction. . Define e Ω ′ = K ∪ Ω ′ and e Ω = e Ω ′ × R n − m . It is proved in [17, Theorem 20] that N ( p, , e Ω ′ ) > N ( p, , K ), and there exists a minimizer e U of the quotient (8) in e Ω ′ . Then e U is a positive weak solution of the equation − ∆ ( y ) p e U = N − p ( p, , e Ω ′ ) e U p − | y | p in e Ω ′ , and thus, − ∆ p e U = N − p ( p, , e Ω ′ ) e U p − | y | p in e Ω . As in Theorem 2.1, this implies N ( p, , e Ω) ≥ N ( p, , e Ω ′ ) > N ( p, , K ) = N ( p, , Ω )(the last equality is due to Theorem 2.1).Thus, there exists u ∈ C ∞ ( e Ω) such that k| y | − u k p, e Ω > N ( p, , Ω ) · k∇ u k p, e Ω . Thismeans N ( p, , Ω) > N ( p, , Ω ) if L is sufficiently large, and the statement follows by theconcentration-compactness principle. (cid:3) In what follows we need some estimates for the solution of the extremal problem (8) for p = 2 in the half-space. For the sake of brevity, we denote q = 2 ∗ σ = 2 nn − σ ; µ q (Ω) = N − (2 , σ, Ω); µ q = µ q ( R n + ) . φ we denote a minimizer of the problem (8) for p = 2 in Ω = R n + . Without loss ofgenerality we can assume k| y | σ − φ k q, R n + = 1. Then φ is a weak solution of the Dirichletproblem − ∆ u = µ q · u q − | y | q (1 − σ ) in R n + , u (cid:12)(cid:12) x n =0 = 0 . (10) Proposition
The function φ satisfies the following relations: φ ( x ) ∼ Cx n , |∇ φ ( x ) | ∼ C, x →
0; (11) φ ( x ) ∼ Cx n | x | n , |∇ φ ( x ) | ≍ C | x | n , x → ∞ . (12) Proof . First, we claim that φ ∈ C γ loc ( R n + ). Indeed, the standard elliptic theory, see, e.g.,[9], provides φ ∈ C ( R n + \ P ). Estimates in the neighborhood of P can be obtained usingelliptic theory in domains with edges, see, e.g., [14]. Note that the property φ ∈ C ( R n + )was proved also in [8, Appendix].Further, the Hopf lemma gives φ x n (cid:12)(cid:12) x n =0 >
0, and (11) follows.Finally, the relations (12) follow from (11). Indeed, the direct computation shows thatthe image of φ under the Kelvin transform is also a solution of the problem (10) while (11)turns into (12). (cid:3) We assume that in a neighborhood of the set
P ∩ ∂ Ω the boundary is of class C ; outside thisneighborhood we impose no assumptions on ∂ Ω. Suppose there exists a point x ∈ P ∩ ∂ Ω(without loss of generality, x = 0) satisfying the properties listed below.Let us introduce local Cartesian coordinates with y ′ = ( y , . . . , y m ) in the tangent planeand the axis Oy directed into Ω. Then in a neighborhood of the origin ∂ Ω is given byequation y = F ( y ′ ; z ). It is evident that F ∈ C and F ( y ′ ; z ) = o ( | y ′ | + | z | ). Moreover, theassumption P ∩
Ω = ∅ implies F (0; z ) ≥ ∂ Ω is average concave in a neighborhood of the origin (see [3]), if forsufficiently small ρ f ( ρ ) := − Z S n − ρ F ( y ′ ; z ) d S ρ ( y ′ , z ) < f ( r ; t ) := − Z S m − r − Z S n − m − t F ( y ′ ; z ) d S r ( y ′ ) d S t ( z ) ,f ( ρ ) := − Z S n − ρ |∇ ′ F ( y ′ ; z ) | d S ρ ( y ′ , z ) , ( ∇ ′ stands for the gradient with respect to ( y ′ , z )) and assume that for sufficiently small ρ cos m − ( β ) sin n − m − ( β ) · | f ( ρ cos( β ) , ρ sin( β )) | ≤ C · | f ( ρ ) | , β ∈ [0 , π , (14)9nd lim ρ → f ( ρ ) f ( ρ ) ρ = 0 . (15)We say that ∂ Ω is average concave in P and P ⊥ directions in a neighborhood ofthe origin, if (13) holds for sufficiently small ρ , andΦ( β ) := lim ρ → f ( ρ cos( β ) , ρ sin( β )) f ( ρ ) ≥ , β ∈ [0 , π . (16)Now we can formulate the main result of the second part of our paper. Theorem
Let ∂ Ω be average concave in P and P ⊥ directions in a neighborhood of theorigin, and let the relations (14) and (15) hold. Suppose also that f is regularly varying oforder α ∈ [1 , n + 1[ at the origin. Then for p = 2 and for any < σ < the infimum in (8)is attained. Let us compare our assumptions with those of [8]. If ∂ Ω is smooth and α = 2, then f ( ρ ) ∼ H ρ , f ( r, t ) ∼ H P r + H P ⊥ t , f ( ρ ) ∼ Cρ near the origin (here H = n − Sp ( ∇ ′ F (0)) is the mean curvature of ∂ Ω at the origin;respectively, H P = m − Sp ( ∇ y ′ F (0)) and H P ⊥ = n − m ) Sp ( ∇ z F (0)).Since P ∩
Ω = ∅ , H P ⊥ is always non-negative. Thus, the relations (13) and (16) meanthat H P < H P ⊥ = 0 . (17)The relations (14) and (15) are automatically fulfilled in this case.One can see that (17) is considerably weaker then the assumptions of [8, Theorem 1.1].We underline also that our hypotheses must be fulfilled at some point x ∈ P ∩ ∂ Ω whilethe authors of [8] constrain the curvatures at any point x ∈ P ∩ ∂ Ω. Moreover, we do notrequire even the existence of the mean curvature (if α < α >
Remark The assumption (15) can fulfil even if the main term of the asymptotic expansionof F vanishes under average. For example, it is the case if F ( y ′ ; z ) = y − y . Remark The assumption (14) is used only to ensure the limit passage under integral signand can be easily weakened. However, it cannot be removed at all, and we prefer to give itin a simple form. In turn, the assumption (16) could be weakened if we had in hands moredetailed information on the function φ . Now consider the limit case α = n + 1. In this case we can drop the assumption (16). Theorem
Let ∂ Ω be average concave in a neighborhood of the origin, and let the relations(14) and (15) hold. Suppose also that f is regularly varying of order n + 1 at the origin, and R δ f ( r ) r n +2 d r = −∞ . Then for p = 2 and for any < σ < the infimum in (8) is attained. roof of Theorems 3.1 and 3.2 . Let { v k } be a minimizing sequence for (8).Without loss of generality we can assume k| y | σ − v k k q, Ω = 1 and v k ⇁ v in ˙ W (Ω).Operating as in the proof of Theorem 2.2, we obtain the alternative — either v is a aminimizer of the extremal problem, or v = 0 and || y | σ − v k | q ⇁ δ ( x − b x ) , |∇ v k | ⇁ µ q (Ω) δ ( x − b x ) , b x ∈ P ∩ ∂ Ω(the convergence is understood in the sense of measures on Ω).We claim that in the second case µ q (Ω) ≥ µ q . Indeed, without loss of generality, v k concentrate near the origin. Further, as in the Corollary 2.1 [12], we can assume supports of v k located in arbitrarily small ball. Since F ( y ′ ; z ) = o ( | y ′ | + | z | ) and F (0; z ) ≥
0, this impliessupp( v k ) ⊂ K κ := { x ∈ R n : y > − κ | y ′ |} for any κ >
0. Hence µ q (Ω) ≥ lim κ → µ q ( K κ ) = µ q ( K ) = µ q . Therefore, to prove the statements we need only to produce a function having the quotient(8) less then µ q . Similarly to [3], we construct such function using a suitable dilation and“bending” of the function φ and multiplying it by a cut-off function with small support.The sharp estimates of behavior of φ (Proposition 2.1) provide the desired result underassumptions on ∂ Ω close to optimal.Choose δ such that for | y ′ | + | z | < δ the relation (13) is satisfied and | F ( y ′ ; z ) | ≤ | y ′ | + | z | .Let us introduce the coordinate transformation Θ ε : x ε − ( x − F ( y ′ ; z ) e m ). It isevident that in a neighborhood of the origin Θ ε straightens ∂ Ω; its Jacobian equals ε − n .Also it is easy to see that for r < δ we have B r ε ⊂ Θ ε ( B r ) ⊂ B rε .Let e ϕ ∈ C ∞ ( R n ) be a function, radially symmetric w.r.t. y and z and satisfying 0 ≤ e ϕ ≤ e ϕ ( x ) ≡ e ϕ ( | y | ; | z | ) = (cid:26) , if | y | < δ and | z | < δ ;0 , if | y | > δ or | z | > δ .We introduce the cut-off function ϕ ( x ) = e ϕ (Θ ( x )). Obviously, the function x ϕ (Θ − ε ( x ))is radially symmetric w.r.t. y and z : ϕ (Θ − ε ( x )) = ϕ ( ε y + F ( ε y ′ ; εz ) , ε y ′ ; εz ) = e ϕ ( ε y , ε y ′ ; εz ) = e ϕ ( ε | y | ; ε | z | ) . Now we define the function φ ε ( x ) = ε − ( n − / φ (cid:16) Θ ε ( x ) (cid:17) ϕ ( x ) . It is easy to see that φ ε ∈ o W (Ω), if δ and ε are sufficiently small.In Sections 4–6 we show that Z Ω | φ ε ( x ) | q | y | q (1 − σ ) dx = 1 − A ( ε )(1 + o δ (1) + o ε (1)) , (18) Z Ω |∇ φ ε ( x ) | dx = µ q + A ( ε )(1 + o δ (1) + o ε (1)) − µ q q A ( ε )(1 + o ε (1)) (19)11we recall that o δ (1) is uniform with respect to ε ). For given δ , in these formulas we have,as ε → A ( ε ) ∼ Cε − f ( ε ); (20) A ( ε ) ∼ Cε − f ( ε ) , under assumptions of Theorem 3.1; Cε n R δε f ( r ) r n +2 d r , under assumptions of Theorem 3.2. (21)The relations (20) and (21) imply A ( ε ) = O ( A ( ε )) (in the case α = n + 1 it followsfrom (26)). Therefore, for sufficiently small δ and ε we have, subject to (13), k∇ φ ε k k| y | σ − φ ε k q = µ q + A ( ε )(1 + o δ (1) + o ε (1)) − µ q q A ( ε )(1 + o ε (1)) (cid:16) − A ( ε )(1 + o δ (1) + o ε (1)) (cid:17) /q == µ q + A ( ε )(1 + o δ (1) + o ε (1)) < µ q , and both Theorems follow. (cid:3) We have, using the Taylor expansion, Z Ω | φ ε ( x ) | q | x | q (1 − σ ) dx = Z R n + | φ ( y ; z ) | q | y + ε − F ( εy ′ ; εz ) e m | q (1 − σ ) ϕ q (Θ − ε ( x )) dydz == Z R n + | φ ( y ; z ) | q | y | q (1 − σ ) e ϕ q ( εy ; εz ) · (cid:18) − q (1 − σ ) F ( εy ′ ; εz ) y ε | y | (cid:19) dydz ++ O δ (1) Z R n + | φ ( y ; z ) | q F ( εy ′ ; εz ) ε | y + ξε − F ( εy ′ ; εz ) e m | q (1 − σ )+2 e ϕ q ( εy ; εz ) dydz =: I − I + I (here ξ = ξ ( y, z ) ∈ ]0 , φ is normalized, I ≤
1. On the another hand, the first estimate in (12) gives1 − I = Z R n + φ q ( y ; z ) | y | q (1 − σ ) (cid:16) − e ϕ q ( εy ; εz ) (cid:17) dydz ≤ C Z R n + \ B δ ε | y | − q (1 − σ ) dydz ( | y | + | z | ) n − ≤ C (cid:16) εδ (cid:17) qn . I = q (1 − σ ) ε Z R n + φ q ( y ; z ) y | y | q (1 − σ )+2 e ϕ q ( εy ; εz ) F ( εy ′ ; εz ) dydz =: A ( ε ) . Proposition
Given δ , the function A ( ε ) satisfies (20), as ε → . roof . We claim thatlim ε → ε A ( ε ) f ( ε ) = q (1 − σ ) ω m − ω n − m − ×× ∞ Z ρ n − α π Z cos m − ( β ) sin n − m − ( β )Φ( β ) ∞ Z φ q ( ρ cos( β ) , s ; ρ sin( β )) s ds ( ρ cos ( β ) + s ) q (1 − σ )+22 dβdρ. (22)To prove this we apply the Lebesgue theorem. We have εA ( ε ) q (1 − σ ) f ( ε ) = 1 f ( ε ) Z R n + e ϕ q ( εy ; εz ) φ q ( y ; z ) y | y | q (1 − σ )+2 F ( εy ′ ; εz ) dydz == 1 f ( ε ) ∞ Z ∞ Z ∞ Z e ϕ q ( ε q r + y ; εt ) φ q ( y , r ; t ) y ( r + y ) q (1 − σ )+22 ×× Z S m − r Z S n − m − t F ( εy ′ ; εz ) d S m − r ( y ′ ) d S n − m − t ( z ) dy drdt == ω m − ω n − m − ∞ Z ∞ Z r m − t n − m − f ( εr ; εt ) f ( ε ) ×× ∞ Z e ϕ q ( ε √ r + s ; εt ) φ q ( r, s ; t ) s ds ( r + s ) q (1 − σ )+22 drdt. We can apply the Monotone Convergence Theorem to the interior integral. Further,the regular behavior of f implieslim ε → f ( ερ cos( β ); ερ cos( β )) f ( ε ) = lim ε → f ( ερ cos( β ); ερ cos( β )) f ( ερ ) · lim ε → f ( ερ ) f ( ε ) = Φ( β ) ρ α . Therefore, the assumption on the pointwise convergence is satisfied. Now we producea summable majorant.Due to the estimates (11) and (12), the interior integral is bounded from above by Cχ ( ε ) ( r ; t ) ∞ Z s qσ − ds r + s + t ) qn ≤ Cχ ( ε ) ( r ; t )1 + ( t + r ) q ( n − σ )2 , where χ ( ε ) ( r ; t ) = χ [0 , δε ] ( r ) · χ [0 , δε ] ( t ).Now we pass to the polar coordinates. Using (14), we estimate the integrand by Cχ [0 , δε ] ( ρ ) · f ( ερ ) f ( ε ) · ρ n − ρ q ( n − σ ) . For γ > χ [0 , δε ] ( ρ ) · f ( ερ ) f ( ε ) ≤ χ [0 , ( ρ ) ρ α − γ f ( ερ ) ( ε ρ ) − α + γ f ( ε ) ε − α + γ + χ [1 , δε ] ( ρ ) ρ α + γ f ( ερ ) ( ερ ) − ( α + γ ) f ( ε ) ε − ( α + γ ) . f is RVF of order α , the function f ( τ ) τ − α + γ increases for small τ and the function f ( τ ) τ − ( α + γ ) decreases for small τ . Therefore, we have χ [0 , δε ] ( ρ ) · f ( ερ ) f ( ε ) · ρ n − ρ q ( n − σ ) ≤ C (cid:18) χ [0 , ( ρ ) ρ α + n − − γ + χ [1 , + ∞ [ ( ρ ) ρ α + n − γ − q ( n − σ ) (cid:19) . Since α ≥
1, the majorant is summable at zero if γ is sufficiently small. Since α ≤ n +1,for small γ the second exponent does not exceed2 n − γ − q ( n − σ ) = − γ − nσn − σ < − , and the majorant is summable at infinity. (cid:3)
3. We recall that F (0; z ) ≥
0, and hence, for small δ and ε | y + ξε − F ( εy ′ ; εz ) e m | ≥≥ | y + ξε − F (0; εz ) e m | − ε − | F ( εy ′ ; εz ) − F (0; εz ) | ≥≥ | y | − | y ′ | o δ (1) ≥ | y | . Therefore, (cid:12)(cid:12)(cid:12)(cid:12) I εf ( ε ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z R n + φ q ( y ; z ) | y | q (1 − σ )+2 e ϕ q ( εy ; εz ) F ( εy ′ ; εz ) ε | f ( ε ) | dydz ≤ C Z R n − χ ( ε ) ( | y ′ | ; | z | ) F ( εy ′ ; εz ) ε | f ( ε ) | ∞ Z φ q ( y ′ , s ; z ) ds ( | y ′ | + s ) q (1 − σ )+22 dy ′ dz. Taking into account (11) and (12), we obtain ∞ Z φ q ( y ′ , s ; z ) ds ( | y ′ | + s ) q (1 − σ )+22 ≤ C ∞ Z ( | y ′ | + s ) qσ − ds | y ′ | + | z | + s ) qn ≤ C | y ′ | qσ − | y ′ | + | z | ) qn , and therefore, (cid:12)(cid:12)(cid:12)(cid:12) I εf ( ε ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C δε Z Z S n − ρ | y ′ | qσ − F ( εy ′ ; εz ) ε | f ( ε ) | d S n − ρ ( y ′ , z ) dρ ρ qn . Since qσ − > −
1, it is easy to see that W ( S n − ) is embedded into L ,w ( S n − ) withweight w = | y ′ | qσ − . Thus, using the Poincar´e inequality, we can write − Z S n − ρ | y ′ | qσ − | F ( y ′ ; z ) | d S n − ρ ( y ′ , z ) ≤≤ Cρ qσ − ρ · − Z S n − ρ |∇ ′ F ( y ′ ; z ) | d S n − ρ ( y ′ , z ) + (cid:18) − Z S n − ρ F ( y ′ ; z ) d S n − ρ ( y ′ , z ) (cid:19) ! . (cid:12)(cid:12)(cid:12)(cid:12) I εf ( ε ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C δε Z ( ερ ) f ( ερ ) + f ( ερ ) ε | f ( ε ) | · ρ n − qσ dρ ρ qn == δε Z o ρε (1) f ( ερ ) f ( ε ) · ρ n − qσ dρ ρ qn = o δ (1) , and we arrive at I = A ( ε ) o δ (1).We remark also that qn > n ≥ α −
1. This implies ε qn = A ( ε ) o ε (1).Choosing δ > α < n + 1 We have ( φ ε ) x m = ε − n/ φ y (Θ ε ( x )) ϕ ( x ) + ε − n/ φ (Θ ε ( x )) ϕ x m ( x );while for i = 1( φ ε ) x i = ε − n/ (cid:18) φ y i (Θ ε ( x )) − φ y (Θ ε ( x )) F x i ( y ′ ; z ) (cid:19) ϕ ( x ) + ε − n/ φ (Θ ε ( x )) ϕ x i ( x ) . Hence Z Ω |∇ φ ε ( x ) | dx = Z Ω (cid:20) ε − n ϕ ( x ) | ( ∇ ′ φ )(Θ ε ( x )) | −− ε − n ϕ ( x ) φ y (Θ ε ( x )) (cid:10) ( ∇ ′ φ )(Θ ε ( x )) , ∇ ′ F ( y ′ ; z ) (cid:11) ++ 2 ε − n +1 φ (Θ ε ( x )) ϕ ( x ) (cid:10) ( ∇ ′ φ )(Θ ε ( x )) , ∇ ′ ϕ ( x ) (cid:11) −− ε − n +1 φ y (Θ ε ( x )) φ (Θ ε ( x )) ϕ ( x ) (cid:10) ∇ ′ F ( y ′ ; z ) , ∇ ′ ϕ ( x ) (cid:11) ++ ε − n φ (Θ ε ( x )) |∇ ′ ϕ ( x ) | + ε − n φ y (Θ ε ( x )) ϕ ( x ) |∇ ′ F ( y ′ ; z ) | ++ ε − n ϕ ( x ) φ y (Θ ε ( x )) + ε − n φ (Θ ε ( x )) ϕ x m ( x ) ++ 2 ε − n +1 φ y (Θ ε ( x )) ϕ ( x ) φ (Θ ε ( x )) ϕ x m ( x ) (cid:21) dx =: J − J + · · · + J . J + J = Z R n + e ϕ ( εy ; εz ) |∇ φ ( y ; z ) | dydz = µ q − Z R n + (1 − e ϕ ( εy ; εz )) |∇ φ ( y ; z ) | dydz ;15oreover, the second estimate in (12) gives Z R n + (1 − e ϕ ( εy ; εz )) |∇ φ ( y ; z ) | dydz ≤ C ∞ Z δ ε ζ − − n dζ = C (cid:16) εδ (cid:17) n , whence J + J = µ q + C ( δ ) O ( ε n ) .
2. Integrating by parts we obtain J = Z R n + e ϕ ( εy ; εz ) φ y ( y ; z ) (cid:10) ∇ ′ φ ( y ; z ) , ∇ ′ F ( εy ; εz ) (cid:11) dydz == − ε Z R n + (cid:20) e ϕ ( εy ; εz ) φ y ( y ; z )∆ ′ φ ( y ; z ) + (cid:10) ∇ ′ ( e ϕ )( εy ; εz ) , ∇ ′ φ ( y ; z ) (cid:11) φ y ( y ; z )++ e ϕ ( εy ; εz ) (cid:10) ∇ ′ φ ( y ; z ) , ∇ ′ φ y ( y ; z ) (cid:11)(cid:21) F ( εy ′ ; εz ) dydz. By (10), we obtain J = 2 ε Z R n + e ϕ ( εy ; εz ) (cid:18) φ y y ( y ; z ) + µ q φ q − ( y ; z ) | y | q (1 − σ ) (cid:19) φ y ( y ; z ) F ( εy ′ ; εz ) dydz −− ε Z R n + (cid:10) ∇ ′ ( e ϕ )( εy ; εz ) , ∇ ′ φ ( y ; z ) (cid:11) φ y ( y ; z ) F ( εy ′ ; εz ) dydz −− ε Z R n + e ϕ ( εy ; εz ) ( |∇ ′ φ ( y ; z ) | ) y F ( εy ′ ; εz ) dydz =: H + K + K . Now we integrate the first term by parts. H = 1 ε Z R n + e ϕ ( εy ; εz ) (cid:18) ( φ y ( y ; z )) y + 2 µ q q ( φ q ( y ; z )) y | y | q (1 − σ ) (cid:19) F ( εy ′ ; εz ) dydz == − ε Z R n − e ϕ ( εy ; εz ) φ y (0 , y ′ ; z ) F ( εy ′ ; εz ) dy ′ dz −− ε Z R n + ( e ϕ ( εy ; εz )) y φ y ( y ; z ) F ( εy ′ ; εz ) dydz −− µ q q ε Z R n + ( e ϕ ( εy ; εz )) y φ q ( y ; z ) | y | q (1 − σ ) F ( εy ′ ; εz ) dydz ++ 2 (1 − σ ) µ q ε Z R n + e ϕ ( εy ; εz ) φ q ( y ; z ) | y | q (1 − σ ) y | y | F ( εy ′ ; εz ) dydz =: − A ( ε ) + K + K + K . roposition . Let assumptions of Theorem 3.1 hold. Then, given δ , the function A ( ε ) satisfies (21), as ε → . Proof . We claim thatlim ε → εA ( ε ) f ( ε ) = ω m − ω n − m − ×× ∞ Z ρ n − α π Z cos m − ( β ) sin n − m − ( β )Φ( β ) |∇ φ (0 , ρ cos( β ); ρ sin( β )) | dβdρ. (23)To prove this we apply the Lebesgue theorem. We have, similarly to Proposition 4.1, εA ( ε ) ω m − ω n − m − f ( ε ) = ∞ Z ∞ Z e ϕ ( εr ; εt ) r m − t n − m − f ( εr ; εt ) f ( ε ) |∇ φ (0 , r ; t ) | drdt. (24)Passing to the polar coordinates, we see that the integrand converges to that in (23)for all ρ and β . Now we produce a summable majorant. By (11) and (12), we have |∇ φ (0 , ρ cos( β ); ρ sin( β )) | ≤ C ρ n . Therefore for γ >
0, similarly to Proposition 4.1, we estimate the integrand by Cχ [0 , δε ] ( ρ ) f ( ε ρ ) f ( ε ) ρ n − ρ n ≤ C (cid:16) χ [0 , ( ρ ) ρ α + n − − γ + χ [1 , ∞ [ ( ρ ) ρ α − n − γ (cid:17) . Since α < n + 1, this provides a summable majorant for sufficiently small γ . (cid:3) Now we estimate all remaining terms in J . Since the functions e ϕ and φ are radiallysymmetric w.r.t y ′ and z , integrating by parts in K we have K + K + K = − ε Z R n + F ( εy ′ ; εz ) (cid:20) (cid:10) ∇ ′ ( e ϕ )( εy ; εz ) , ∇ ′ φ ( y ; z ) (cid:11) φ y ( y ; z ) −− ( e ϕ ( εy ; εz )) y |∇ ′ φ ( y ; z ) | + ( e ϕ ( εy ; εz )) y φ y ( y ; z ) (cid:21) dydz == − ε ∞ Z ∞ Z ∞ Z r m − t n − m − f ( εr ; εt ) ×× h (cid:0) ( e ϕ ( εy , εr ; εt )) r φ r + ( e ϕ ( εy , εr ; εt )) t φ t (cid:1) φ y −− ( e ϕ ( εy , εr ; εt )) y (cid:0) φ r + φ t (cid:1) + ( e ϕ ( εy , εr ; εt )) y φ y i dy drdt. | K + K + K | ≤ Cδ Z Z δ ε ≤ √ ρ + y ≤ δε |∇ φ | ρ n − | f ( ερ ) | dy dρ ≤≤ Cδ Z Z δ ε ≤ √ ρ + s ≤ δε ρ n − | f ( ερ ) | ( ρ + s ) n ds dρ = Cε n δ Z Z δ ≤√ r + s ≤ δ r n − | f ( r ) | ( r + s ) n d s d r = C ( δ ) · ε n . In a similar way, K = − µ q q ε ∞ Z ∞ Z ∞ Z r m − t n − m − f ( εr ; εt ) · ( e ϕ ( εy , εr ; εt )) y φ q ( y , r ; t )( r + y ) q (1 − σ )2 dy drdt, and therefore, | K | ≤ Cδ Z Z δ ε ≤ √ ρ + y ≤ δε y qσ ρ n − | f ( ερ ) | ( ρ + y ) qn dy dρ == Cε q ( n − σ ) − n δ Z Z δ ≤√ r + s ≤ δ s qσ r n − | f ( r ) | ( r + s ) qn d s d r = C ( δ ) · ε q ( n − σ ) − n = o ( ε n )(the last relation follows from q ( n − σ ) − n = nσn − σ > K can be estimated in the same way as I in Section 4. Thisgives, as ε → K ∼ µ q q A ( ε ) . Thus, J = − A ( ε ) + C ( δ ) O ( ε n ) + 2 µ q q A ( ε )(1 + o ε (1)) .
3. By the estimate (12), we obtain | J + J | ≤ ε Z R n + φ ( y ; z ) |∇ φ ( y ; z ) | e ϕ ( εy ; εz ) |∇ e ϕ ( εy ; εz ) | dydz ≤≤ C εδ δε Z δ ε ζ − n dζ = C (cid:16) εδ (cid:17) n .
4. Using the previous estimate, we obviously get | J | ≤ ε Z R n + φ ( y ; z ) |∇ φ ( y ; z ) | e ϕ ( εy ; εz ) |∇ e ϕ ( εy ; εz ) | |∇ ′ F ( εy ′ ; εz ) | dydz ≤≤ C ε Z R n + φ ( y ; z ) |∇ φ ( y ; z ) | e ϕ ( εy ; εz ) |∇ e ϕ ( εy ; εz ) | dydz ≤ C (cid:16) εδ (cid:17) n .
18. In a similar way, | J + J | = ε Z R n + φ ( y ; z ) |∇ e ϕ ( εy ; εz ) | dydz ≤ C ε δ δε Z δ ε ζ − n dζ = C (cid:16) εδ (cid:17) n .
6. Finally, relations (12) and (15) imply J = Z R n + φ y ( y ; z ) e ϕ ( εy ; εz ) |∇ ′ F ( εy ′ ; εz ) | dydz ≤≤ C δε Z ρ n − f ( ερ ) ∞ Z dy (1 + ρ + y ) n dρ = ≤ C δε Z ρ n − f ( ερ )(1 + ρ ) n − / dρ = o δ (1) ε δε Z ρ n − | f ( ερ ) | (1 + ρ ) n − / dρ. The last integral can be estimated in the same way as in Proposition 5.1. This gives J = o δ (1) A ( ε ) . We remark also that ε n = A ( ε ) o ε (1).Choosing δ > α = n + 1 We underline that the assumption α < n + 1 was used in the previous section only in theproof of Proposition 5.1. Also the assumption (16) was used only to ensure the positivity ofthe integral in (23). So, we need only to prove the following fact.
Proposition
Let assumptions of Theorem 3.2 hold. Then, given δ , the function A ( ε ) satisfies (21), as ε → . Proof . By (12), there exists
M > |∇ φ (0 , r ; t ) | = M + o ρ (1) ρ n , ρ = √ r + t → ∞ . (25)We split the integral (24) into three parts: A ( ε ) ω m − ω n − m − ε n = (cid:18) Z Z √ r + t ≤ R + Z Z R ≤√ r + t ≤ δ ε + Z Z δ ε ≤√ r + t ≤ δε (cid:19) e ϕ ( εr ; εt ) ×× r m − t n − m − f ( εr ; εt ) ε n +1 |∇ φ (0 , r ; t ) | drdt =: L + L + L . R → ∞ , L = ( M + o R (1)) ×× δ ε Z R π Z cos m − ( β ) sin n − m − ( β ) f ( ερ cos( β ); ερ sin( β )) ε n +1 ρ n +2 dβ dρ == ( M + o R (1)) ω n − ω m − ω n − m − δ ε Z R f ( ερ ) ε n +1 ρ n +2 dρ = ( M + o R (1)) ω n − ω m − ω n − m − δ/ Z Rε f ( r ) r n +2 d r . Further, the assumption (14) implies (cid:12)(cid:12)(cid:12)(cid:12) ε n +1 L f ( ε ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C R Z f ( ερ ) f ( ε ) ρ n − dρ. For given R we can pass to the limit under the integral sign. This provides L = O (cid:0) f ( ε ) ε n +1 (cid:1) ,as ε → R δ f ( r ) r n +2 d r implies that for arbitrary large N we have, as ε is sufficiently small, ε n +1 f ( ε ) δ/ Z Rε f ( r ) r n +2 d r ≥ ε n +1 f ( ε ) Nε Z Rε f ( r ) r n +2 d r = N Z R f ( ερ ) f ( ε ) ρ n +1 dρρ = ln( N/R ) · (1 + o ε (1)) , (26)and thus, L = o ( L ), as ε → ε → | L | ≤ C δε Z δ ε f ( ερ ) ε n +1 ρ n +2 dρ = C δ Z δ/ f ( r ) r n +2 d r = C ( δ ) = o ( L ) . It remains to note that for given R and δ δ/ Z Rε f ( r ) r n +2 d r = δ Z ε f ( r ) r n +2 d r + O (1) ∼ δ Z ε f ( r ) r n +2 d r , ε → , and we arrive at A ( ε ) ∼ M ω n − ε n δ Z ε f ( r ) r n +2 d r . (cid:3) eferences [1] J.E. Brothers & W.P. Ziemer, Minimal rearrangements of Sobolev functions , J. reineangew. Math. (1988), 153–179.[2] A.V. Demyanov & A.I. Nazarov,
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