Sharp Hardy inequalities in an exterior of a ball
aa r X i v : . [ m a t h . A P ] S e p Sharp Hardy inequalities in an exterior of a ball
Nikolai Kutev ∗ Tsviatko Rangelov ∗ Abstract
New Hardy type inequalities in sectorial area and as a limit in an exterior of a ballare proved. Sharpness of the inequalities is shown as well.
Keywords:
Hardy inequalities, sharp estimates.
The classical Hardy inequality, proved in Hardy [7, 8] states Z ∞ | u ′ ( x ) | p x α dx ≥ (cid:18) p − − αp (cid:19) p Z ∞ x − p + α | u ( x ) | p dx (1)where 1 < p < ∞ , α < p − u ( x ) is absolutely continuous on [0 , ∞ ), u (0) = 0.There are several generalizations of (1) in multidimensional case mostly in boundeddomains, see Davies [4], Ghoussoub and Moradifam [6], Balinsky et al. [3], Kutev andRangelov [9] and the literature therein. For unbounded domains, in the exterior of aboll, only few generalizations of (1) are reported in Wang and Zhu [10], Adimurthi et al.[1], Avkhadiev and Laptev [2].For example, in Theorem 1.1 in Wang and Zhu [10] for all function u ∈ D , a ( R n ),where D , a ( R n ) is the weighted Sobolev space - the comletion of C ∞ ( R n ) with the norm Z R n | x | − a | u | dx , the following inequality for a < n −
22 is proved Z B c | x | − a |∇ u | dx ≥ (cid:18) n − − a (cid:19) Z B c | u | | x | ( a + 1) dx + n − − a Z ∂B c u , (2)where B c = {| x | > } .Let us mention that in Wang and Zhu [10] Hardy inequalities with weights in un-bounded domains Ω ⊂ R n , / ∈ ∂ Ω are also considered, see Theorem 1.3 and Remark1.5.In Adimurthi et al. [1], Corollary, for n ≥ u ∈ C ∞ ( B c ) the limiting case of theCaffarelli–Kohn–Nirenberg inequality Z B c |∇ u | dx ≥ (cid:18) n − (cid:19) Z B c u | x | dx + C n ( a ) "Z B c X n − n − (cid:18) a, | x | (cid:19) u nn − dx n − n , (3) ∗ Institute of Mathematics and Informatics, Bulgarian Academe of Sciences, 1113, Sofia, BulgariaCorresponding author: T. Rangelov, [email protected]
1s proved, where X ( a, s ) = a − ln s ) − , a > , < s ≤ C n ( a ) is givenexplicitly.Finally, in Corollary 1 and Remark 1 in Avkhadiev and Laptev [2] the following Hardyinequality is proved for n ≥ u ∈ W , ( B cr ) Z B cr |∇ u | dx ≥ (cid:18) n − (cid:19) Z B cr u | x | dx + 14 Z B cr (cid:18) | x − r | − | x | (cid:19) u dx. (4)At the end of the paper we compare the inequalities (2), (3), (4) with our results.The aim of the present work is to derive new Hardy inequalities in the exterior of aball. There are listed also functions for which these inequalities are sharp, i.e., inequalitieswith an optimal constant of the leading term become equations. We start with Hardy inequalities in sectorial area B R \ B r where B R , B r are balls centeredat zero, 0 < r < R . Let 1 < p , p ′ = pp − ≤ n and denote m = p − np − p − np p ′ .For functions u such that 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 < ∞ let us define two sets M ( r, R ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R m − ˆ R m m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − p Z ∂B ˆ R | u | p dσ → , 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) − n Z ∂B ˆ R | u | n dσ → , ˆ R → R − , m = 0 M ( r, R ) = (cid:12)(cid:12)(cid:12)(cid:12) ˆ r m − r m m (cid:12)(cid:12)(cid:12)(cid:12) − p Z ∂B ˆ r | u | p dσ → , ˆ r → r + 0 , m = 0 , (cid:12)(cid:12)(cid:12)(cid:12) ln ˆ rr (cid:12)(cid:12)(cid:12)(cid:12) − n Z ∂B ˆ r | u | n dσ → , ˆ r → r + 0 , m = 0where h , i is a scalar product in R n .Let functions ψ j be solutions of the problems: − ∆ p ψ = 0 , in B R \ ¯ B r , ψ | ∂B R = 0 , ψ | ∂B r = 1 , − ∆ p ψ = 0 , in B R \ ¯ B r , ψ | ∂B R = 1 , ψ | ∂B r = 0 . Their explicit form is: ψ ( x ) = R m − | x | m R m − r m , m = 0 , ln R | x | ln Rr , m = 0 , ψ ( x ) = | x | m − r m R m − r m , m = 0 , ln | x | r ln Rr , m = 0 .
2e can define vector functions f i as f i = (cid:12)(cid:12)(cid:12)(cid:12) ∇ ψ i ψ i (cid:12)(cid:12)(cid:12)(cid:12) p − ∇ ψ i ψ i . in B R \ ¯ B r and L i ( u ) = Z B R \ B r (cid:12)(cid:12)(cid:12)(cid:12) h∇ ψ i , ∇ u i|∇ ψ i | (cid:12)(cid:12)(cid:12)(cid:12) p dx = 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,K i ( u ) = Z B R \ B r (cid:12)(cid:12)(cid:12)(cid:12) ∇ ψ i ψ i (cid:12)(cid:12)(cid:12)(cid:12) p | u | p dx = Z B R \ B r | f i | p ′ | u | p dx,K i ( u ) = Z ∂ ( B R \ B r ) (cid:12)(cid:12)(cid:12)(cid:12) ∇ ψ i ψ i (cid:12)(cid:12)(cid:12)(cid:12) p − (cid:28) ∇ ψ i ψ i , ν (cid:29) | u | p ds = Z ∂ ( B R \ B r ) h f i , ν i| u | p dx. (5)Here ν is the outward normal to B R \ B r .The following Theorem takes place. Theorem 2.1.
For every u ∈ M i ( r, R ) we have L i ( u ) ≥ (cid:18) p (cid:19) p [ K i ( u ) + ( p − K i ( u )] p ( K i ( u )) p − = K i ( u ) . (6) where ν is the outward normal to B R \ B r .Proof. We follow the proof of Proposition 1 in Fabricant et al. [5]. Since Z B ˆ R \ B ˆ r h f i , ∇| u | p i dx = p Z B ˆ R \ B ˆ r | u | p − u h f i , ∇ u i dx, (7)where r < ˆ r < ˆ R < R . Then applying H¨older inequality on the rhs of (7) with h x, ∇ u i| x | and | f i || u | p − u as factors of the integrand we get Z B ˆ R \ B ˆ r h f i , ∇| u | p i dx ≤ p Z B ˆ R \ B ˆ 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 × Z B ˆ R \ B ˆ r | f i | p ′ | u | p dx ! /p ′ . (8)Rising to p power both sides of (8) it follows that Z B ˆ R \ B ˆ 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:12)(cid:12)(cid:12) p R B ˆ R \ B ˆ r h f i , ∇| u | p i dx (cid:12)(cid:12)(cid:12) p (cid:16)R B ˆ R \ B ˆ r | f i | p ′ | u | p dx (cid:17) p − . (9)Integrating by parts the numerator of the right hand side of (9) we get1 p Z B ˆ R \ B ˆ r h f i , ∇| u | p i dx = 1 p Z ∂B ˆ R ∪ ∂B ˆ r h f i , ν i| u | p dS − p Z B ˆ R \ B ˆ r div f i | u | p dx = 1 p Z ∂B ˆ R ∪ ∂B ˆ r h f i , ν i| u | p dS + (cid:18) p − p (cid:19) Z B ˆ R \ B ˆ r | f | p ′ | u | p dx → p (( p − K i + K i ) , when ˆ R → R − , ˆ r → r + 0 . (10)3ote that Z ∂B R ∪ ∂B r h f i , ν i| u | p dS ≥ u ∈ M i ( r, R ), since ν | ∂B R = x | x | | ∂B R , ν | ∂B r = − x | x | | ∂B r . From (9) and (10) we obtain (6) since − div f i = ( p − | f i | p ′ . B cr = R n \ ¯ B r Let us introduce functions u such that Z B cr (cid:12)(cid:12)(cid:12)(cid:12) < x, ∇ u > | x | (cid:12)(cid:12)(cid:12)(cid:12) p dx < ∞ , Z B cr | u | p | x | ( n − p ′ dx < ∞ and define two sets M ( r, ∞ ) = R − n Z ∂B R | u | p dσ → , R → ∞ , m ≥ ,R − p Z ∂B R | u | p dσ → , R → ∞ , m < .M ( r, ∞ ) = (cid:12)(cid:12)(cid:12)(cid:12) ˆ r m − r m m (cid:12)(cid:12)(cid:12)(cid:12) − p Z ∂B ˆ r | u | p dσ → , ˆ r → r + 0 , m = 0 , (cid:18) ln ˆ rr (cid:19) − p Z ∂B ˆ r | u | p dσ → , ˆ r → r + 0 , m = 0 . In a similar way we can prove Theorem 2.1, replacing B R \ ¯ B r with B cr = R n \ ¯ B r and ∂ ( B R \ ¯ B r ) with ∂B cr = ∂B r and L i ( u ) , K I ( u ) , K i ( u ) define in (5) for R → ∞ . Theinequalities below for functions of M i ( r, ∞ ) , i = 1 , R → ∞ . Proposition 3.1.
For every u ∈ M ( r, ∞ ) the following inequalities hold:(i) Z B cr | u | p | x | ( n − p ′ dx ! p ′ Z B cr (cid:12)(cid:12)(cid:12)(cid:12) < x, ∇ u > | x | (cid:12)(cid:12)(cid:12)(cid:12) p dx ! p ≥ p r − n Z ∂B r | u | p dS, f or m > . (11) With function u α ( x ) = e − α | x | m , α > inequality (11) becomes equality.(ii) For m < we get: Z B cr (cid:12)(cid:12)(cid:12)(cid:12) < x, ∇ u > | x | (cid:12)(cid:12)(cid:12)(cid:12) p dx ! p ≥ | m | p ′ Z B cr | u | p | x | p dx ! p + 1 p r − p Z ∂B r | u | p dS Z B cr | u | p | x | p dx ! − p ′ , f or m < . (12) With function u k ( x ) = | x | km , k > p ′ inequality (12) becomes an equality. iii) Z B cr | u | n | x | n dx ! n − n Z B cr (cid:12)(cid:12)(cid:12)(cid:12) < x, ∇ u > | x | (cid:12)(cid:12)(cid:12)(cid:12) n dx ! n ≥ n r − n Z ∂B r | u | n dS, f or m = 0 . (13) With function u q ( x ) = | x | q , q < inequality (13) becomes equality.Proof. For m = 0 the inequality (6) has the form 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 ! p ≥ p ′ Z B R \ B r | u | p | x | ( n − p ′ (cid:12)(cid:12)(cid:12) R m −| x | m m (cid:12)(cid:12)(cid:12) p dx p + 1 p r − n (cid:12)(cid:12)(cid:12)(cid:12) R m − r m m (cid:12)(cid:12)(cid:12)(cid:12) − p Z ∂B r | u | p dS Z B R \ B r | u | p | x | ( n − p ′ (cid:12)(cid:12)(cid:12) R m −| x | m m (cid:12)(cid:12)(cid:12) p dx − p ′ . (14)Analogously, for m = 0, i. e. p = n the inequality (6) becomes 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) ln R | x | (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) ln R | x | (cid:12)(cid:12)(cid:12) n dx − nn . (15)(i) For m >
0, after the limit R → ∞ in (14) we obtain1 p ′ Z B R \ B r | u | p | x | ( n − p ′ (cid:12)(cid:12)(cid:12) R m −| x | m m (cid:12)(cid:12)(cid:12) p dx p → R →∞ p r − n (cid:12)(cid:12)(cid:12)(cid:12) R m − r m m (cid:12)(cid:12)(cid:12)(cid:12) − p Z ∂B r | u | p dS Z B R \ B r | u | p | x | ( n − p ′ (cid:12)(cid:12)(cid:12) R m −| x | m m (cid:12)(cid:12)(cid:12) p dx − p ′ → R →∞ p r − n Z ∂B r | u | p dS Z B cr | u | p | x | ( n − p ′ dx ! − p ′ . and hence (11) holds.Let us check that with the function u α ( x ) = e − α | x | m , α > I α = Z B cr e − αp | x | m | x | ( n − p ′ dx ! p ′ = (cid:18)Z S Z ∞ r e − αpρ m ρ − ( n − p ′ ρ n − dρdS (cid:19) p ′ = | S | p ′ (cid:18) m Z ∞ r e − αpρ m dρ m (cid:19) p ′ = | S | p ′ αpm ) p ′ e − α ( p − r m ,I α = Z B cr (cid:12)(cid:12)(cid:12)(cid:12) < x, ∇ u > | x | (cid:12)(cid:12)(cid:12)(cid:12) p dx ! p = (cid:18)Z S Z ∞ r e − αpρ m ( αm ) p ρ − ( n − p ′ ρ n − dρdS (cid:19) p = | S | p (cid:18) ( αm ) p mαp Z ∞ r e − αpρ m dρ m (cid:19) p = | S | p αm ( αpm ) p e − αr m ,I α = 1 p r − n Z ∂B r | u α | p dS = 1 p r − n Z S e − αpr m r n − dS = | S | p e − αpr m , and we get the equality I α I α = I α .(ii) For m <
0, after the limit R → ∞ in (14) we obtain (12). Since k > p ′ it followsthat u k ( x ) ∈ M ( r, ∞ ). Moreover, inequality (12) becomes equality. Indeed, I k = Z B cr (cid:12)(cid:12)(cid:12)(cid:12) < x, ∇ u k > | x | (cid:12)(cid:12)(cid:12)(cid:12) p dx ! p = Z B cr (cid:12)(cid:12)(cid:12) km | x | km − (cid:12)(cid:12)(cid:12) p dx ! p = | km | (cid:18) | S | Z ∞ r ρ ( km − p ρ n − dρ (cid:19) p = | km | | S | r ( km − p + n | ( km − p + n | ! p ,I k = | m | p ′ Z B cr | u k | p | x | p dx ! p = | m | p ′ (cid:18) | S | Z ∞ r ρ ( km − p + n − dρ (cid:19) p = | m | p ′ | S | r ( km − p + n | ( km − p + n | ! p ,I k = 1 p r − p Z ∂B r | u k | p dS Z B cr | u k | p | x | p dx ! − p ′ = 1 p r − p | S | r kmp + n − | S | r ( km − p + n ( km − p + n ! − p ′ = 1 p (cid:16) | S | r | ( km − p + n | (cid:17) p ( | ( km − p + n | ) p ′ . Since | km || ( km − p + n | = | m | p ′ | ( km − p + n | + 1 p
6e get the equality I k = I k + I k .(iii) For m = 0, after the limit R → ∞ in (15), since n − n Z B R \ B r | u | n | x | n (cid:12)(cid:12)(cid:12) ln R | x | (cid:12)(cid:12)(cid:12) n dx n → R →∞ 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) ln R | x | (cid:12)(cid:12)(cid:12) n dx − nn = 1 n r − n ln − n R (cid:18) − ln r ln R (cid:19) − n Z ∂B r | u | n dS − n R Z B R \ B r | u | n | x | n (cid:12)(cid:12)(cid:12) − ln | x | ln R (cid:12)(cid:12)(cid:12) n dx − nn → R →∞ n r − n Z ∂B r | u | n dS Z B cr | u | n | x | n dx ! − nn we obtain (13). Let us check that with function u q ( x ) = | x | q , q < I q = Z B cr | u q | n | x | n dx ! n − n = (cid:18) | S | Z ∞ r ρ ( q − n ρ n − dρ (cid:19) n − n = | S | n − n ( n | q | ) − n − n r ( n − q ,I q = Z B cr (cid:12)(cid:12)(cid:12)(cid:12) < x, ∇ u q > | x | (cid:12)(cid:12)(cid:12)(cid:12) n dx ! n = (cid:18) | S | Z ∞ r | q | n ρ ( q − n ρ n − dρ (cid:19) n = | S | n | q | n − n n − n r q ,I q = 1 n r − n Z ∂B r | u q | n dS = 1 n r − n | S | r nq r n − = | S | n r qn , and we get the equality I q I q = I q . Proposition 3.2.
For every u ∈ M ( r, ∞ ) the following inequalities hold:(i) Z B cr (cid:12)(cid:12)(cid:12)(cid:12) < x, ∇ u > | x | (cid:12)(cid:12)(cid:12)(cid:12) p dx ! p ≥ mp ′ Z B cr | u | p | x | ( n − p ′ | r m − | x | m | p dx ! p + 1 p limsup R →∞ (cid:20) R − p Z ∂B R | u | p dS (cid:21) Z B cr | u | p | x | ( n − p ′ || x | m − r m | p dx ! − p ′ ,f or m > . (16)7 ith function u ε ( x ) = | x | − m (1 − ε ) p ′ ( | x | m − r m ) (1+ ε ) p ′ , < ε < inequality (16) is ε -sharp, i.e. (cid:18) mp ′ (cid:19) p ≤ L ( u ε ) h ( K ( u ε )) p + K ( u ε ) ( K ( u ε )) − p ′ i p ≤ L ( u ε ) K ( u ε ) ≤ (cid:18) mp ′ (cid:19) p (1 + ε ) p . (17) (ii) Z B cr (cid:12)(cid:12)(cid:12)(cid:12) < x, ∇ u > | x | (cid:12)(cid:12)(cid:12)(cid:12) p dx ! p ≥ | m | p ′ Z B cr | u | p | x | ( n − p ′ | r m − | x | m | p dx ! p + r n − p p limsup R →∞ (cid:20) R − n Z ∂B R | u | p dS (cid:21) Z B cr | u | p | x | ( n − p ′ || x | m − r m | p dx ! − p ′ ,f or m < . (18) With function u s ( x ) = ( r m − | x | m ) s , s > p ′ inequality (18) becomes equality.(iii) Z B cr (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 cr | u | n | x | n (cid:12)(cid:12)(cid:12)(cid:12) ln | x | r (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 cr | u | n | x | n (cid:12)(cid:12)(cid:12)(cid:12) ln | x | r (cid:12)(cid:12)(cid:12)(cid:12) n dx − nn ,f or m = 0 . (19) For function u η ( x ) = (cid:18) ln | x | r (cid:19) n − n (1+ η ) M n − n (1 − η ) , r < | x | < M (cid:18) ln | x | r (cid:19) n − n (1+ η ) | x | n − n (1 − η ) , M < | x | , where < η < , inequality (19) is η -sharp, i.e. (cid:18) n − n (cid:19) n ≤ L ( u η ) K ( u η ) ≤ (cid:18) n − n (cid:19) n (1 + η ) n . (20)8 roof. For m = 0 inequality (6) has the form 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 ! p ≥ | m | p ′ 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 ′ . (21)while for m = 0 (6) becomes 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) ln | x | r (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) ln | x | r (cid:12)(cid:12)(cid:12) n dx − n ′ . (22)(i) If m > R → ∞ in (21) we obtain1 p limsup R →∞ (cid:20) R − n | R m − r m | − p Z ∂B R | u | p dS (cid:21) = 1 p limsup R →∞ " R − n R n − p (cid:12)(cid:12)(cid:12)(cid:12) − r m R m (cid:12)(cid:12)(cid:12)(cid:12) − p Z ∂B R | u | p dS = 1 p limsup R →∞ R − p Z ∂B R | u | p dS which proves (16).For the function u ε ( x ) = | x | − m (1 − ε ) p ′ ( | x | m − r m ) (1+ ε ) p ′ , 0 < ε < u ε ( x ) ∈ M ( r, ∞ )9imple computation give us I ε = L ( u ε ) = Z B cr (cid:12)(cid:12)(cid:12)(cid:12) < x, ∇ u ε > | x | (cid:12)(cid:12)(cid:12)(cid:12) p dx = (cid:18) mp ′ (cid:19) p Z B cr | x | − m (1 − ε )( p − − p ( | x | m − r m ) (1+ ε ( p − − p × [(1 + ε ) | x | m − (1 − ε ) ( | x | m − r m )] p dx = (cid:18) mp ′ (cid:19) p | S | Z ∞ r ( ρ m − r m ) (1+ ε )( p − − p × [(1 + ε ) ρ m − (1 − ε )( ρ m − r m )] p ρ − m (1 − ε )( p − − p + n − dρ ≤ (cid:18) mp ′ (cid:19) p | S | (1 + ε ) p Z ∞ r ( ρ m − r m ) (1+ ε )( p − − p ρ − m (1 − ε )( p − − p + n − mp dρ = (cid:18) mp ′ (cid:19) p | S | (1 − ε ) p Z ∞ r ( ρ m − r m ) (1+ ε )( p − − p ρ − m (1+ ε )( p − − n − p − dρ, because n − − p + mp = − n − p − I ε = K ( u ε ) = Z B cr | x | − m (1 − ε )( p − − ( n − p ′ ( | x | m − r m ) (1+ ε )( p − − p dx = | S | Z ∞ r ( ρ m − r m ) (1+ ε )( p − − p ρ − m (1 − ε )( p − − n − p − dρ. Thus (17) follows immediately from expressions of I ε and I ε . (ii) When m <
0, after thelimit R → ∞ in (21) we get1 p limsup R →∞ (cid:20) R − n | R m − r m | − p Z ∂B R | u | p dS (cid:21) = 1 p limsup R →∞ " r n − p R − n (cid:12)(cid:12)(cid:12)(cid:12) R m r m − (cid:12)(cid:12)(cid:12)(cid:12) − p Z ∂B R | u | p dS = r n − p p limsup R →∞ R − n Z ∂B R | u | p dS which proves (18). 10or the function u s ( x ) = ( r m − | x | m ) s , s > p ′ we have the identities I s = Z B cr (cid:12)(cid:12)(cid:12)(cid:12) < x, ∇ u s > | x | (cid:12)(cid:12)(cid:12)(cid:12) p dx ! p = (cid:18) s p | m | p | S | Z ∞ r ρ p ( m − n − ( r m − ρ m ) ( s − p dρ (cid:19) p = s | m || S | p (cid:18) − m Z ∞ r ( r m − ρ m ) ( s − p d ( r m − ρ m ) (cid:19) p , = s | m | p ′ | S | p r mp [( s − p +1] [( s − p + 1] − p because p ( m −
1) + n − m − I s = Z B cr | u s | p | x | ( n − p ′ | r m − | x | m | p dx = | S | Z ∞ r ρ ( n − − p ′ ) | r m − ρ m | ( s − p dρ = | S || m | r m [( s − p +1] ( s − p + 1 ,I s = r n − p p limsup R →∞ (cid:20) R − n Z ∂B R | r m − R m | sp dS (cid:21) = r n − p p limsup R →∞ | R m − r m | sp | S | = | S | p r n − p + mps . Simple computation gives us the equality | m | p ′ ( I s ) p + I s ( I s ) − p ′ = | S || m | p ′ r mp [( s − p +1] p ′ [( s − p + 1] p + | S | r n − p + mps p (cid:18) | S || m | (cid:19) − p ′ " r m [( s +1) p +1] ( s − p + 1 − p ′ = 1 p | m | p ′ | S | p r mp [( s − p +1] (cid:18) p − s − p + 1] p + [( s − p + 1] p ′ (cid:19) = s | m | p ′ | S | p r mp [( s − p +1] [( s − p + 1] − p = I s because n − p + mps − mp ′ [( s − p + 1] = mp [( s − p + 1](iii) After the limit R → ∞ in (22) we get (19).The function u η ( x ) belongs to M ( r, ∞ ) for m = 0. Moreover, for r < | x | < M wehave the equalities I η = L ( u η ) = M ( n − − η ) (cid:16) η (cid:17) n (cid:18) n − n (cid:19) n | S | Z Mr (cid:16) ln ρr (cid:17) ( n − ( η ) − n ρ − dρ,I η = K ( u η ) = M ( n − η ) | S | Z Mr (cid:16) ln ρr (cid:17) ( n − ( η ) − n ρ − dρ, L ( u η ) K ( u η ) = (cid:18) n − n (cid:19) n (cid:16) η (cid:17) n ≤ (cid:18) n − n (cid:19) n (1 + η ) n . Tedious calculations give us for | x | ≥ M the identities I η = L ( u η ) = (cid:18) n − n (cid:19) n | S | Z ∞ M ρ ( n − − η ) − (cid:16) ln ρr (cid:17) ( n − ( η ) − n × (cid:20)(cid:16) − η (cid:17) ρ ln ρr + (cid:16) η (cid:17)(cid:21) n dρI η = K ( u η ) = | S | Z Mr ρ ( n − − η − (cid:16) ln ρr (cid:17) ( n − ( η ) − n , If M is sufficiently large, i.e.,1 M ln Mr < η − η and M > er, then (cid:16) − η (cid:17) ρ ln ρr + 1 + η ≤ (cid:16) − η (cid:17) η − η + 1 + η η, because the function h ( ρ ) = 1 ρ ln ρr is monotone decreasing for ρ > er . Thus (20) followsfrom expressions of I η and I η above for | x | ≥ r . B cr for u ∈ W ,p ( B cr ) For m <
0, i.e. p < n , we can combine inequalities (14) and (12) for functions u ∈ M ( r, ∞ ),where M ( r, ∞ ) = u ∈ W ,p ( B cr ) , (cid:18) ˆ r m − r m m (cid:19) − p Z ∂B ˆ r | u | p dx → , ˆ r → r + 0 ,R − p Z ∂B R | u | p dx → , for R → ∞ . For r < γ < ∞ , γ = 2 | m | r , we define L ( u ) = Z B γ \ 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, L ( u ) = Z B cγ (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 ) = | m | p Z B γ \ B r | u | p | x | ( n − p ′ ( | x | m − r m ) p dx, K ( u ) = | m | p Z B cγ | u | p | x | p dx,K ( u ) = | m | p − γ − p Z ∂B γ | u | p dσ = K ( u ) = | m | p − γ − p Z ∂B γ | u | p dσ = K . roposition 4.1. If γ = 2 / | m | r then for every u ∈ M ( r, ∞ ) the inequality L ( u ) = X L j ( u ) = Z B cr (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 X [ K j ( u ) + ( p − K j ( u )] p ( K j ( u )) p − = K ( u ) , (23) holds in B cr .The inequality (23) becomes an equality for functions u β ( x ) ∈ M ( r, ∞ ) , β > /p ′ where u β ( x ) = ( r m − | x | m ) β x ∈ B γ \ B r , | x | mβ , x ∈ B cγ . Proof.
As in Theorem 2.1 and Proposition 3.1 ii) we get L ( u ) ≥ K ( u ) in B γ \ B r ,L ( u ) ≥ K ( u ) in B cγ (24)where K j ( u ) = [ K j ( u ) + ( p − K j ( u )] p ( K j ( u )) p − . With the choice of γ = 2 / | m | r , so γ p = γ ( n − p ′ ( r m − γ m ) p , we have continuous kernel for K j on ∂B γ .Adding inequalities in (24) we get for u ∈ M ( r, ∞ ) L ( u ) = X L j ( u ) ≥ X K j ( u ).Hance from the Joung inequality we obtain (23)Using (12) in B cγ and (15) in B γ \ r it follows that the inequality (23) becomes equalityfor functions u β ( x ) , β > p ′ .Proposition 4.1 gives sharp Hardy inequality in an exterior of a ball B cr for functions u ∈ W ,p ( B cr ), p < n .We will illustrate Proposition 3.2 ii) and Proposition 4.1 in the following examples. Example 4.1.
For p = 2, n ≥ r = 1, a = 0 and m = 2 − n < Z B c |∇ u | dx ≥ (cid:18) n − (cid:19) Z B c | u | | x | dx + n − Z ∂B c u + 14 Z ∂B c u Z B c | u | | x | ! − , (25)holds. Note that (25) has an additional term in the right hand side in comparison with (2).Moreover, for the functions u k ( x ) = | x | k (2 − n ) , k > xample 4.2. For p = 2 , n ≥ , m = 2 − n < γ = 2 n − r , and every function u ∈ W , ( B cr ) the Hardy inequality (24) becomes Z B cr (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) n − (cid:19) Z B cr u | x | dx +2 (cid:18) n − (cid:19) Z B γ \ B r (cid:12)(cid:12) xr (cid:12)(cid:12) n − (cid:18) − (cid:12)(cid:12)(cid:12) xγ (cid:12)(cid:12)(cid:12) n − (cid:19)(cid:16)(cid:12)(cid:12) xr (cid:12)(cid:12) n − − (cid:17) u dx + 2 − n − r ( n − Z ∂B γ u dS + 2 − n − r (cid:18) n − (cid:19) Z ∂B γ u dS ! × Z B γ \ B r u | x | (cid:16)(cid:12)(cid:12) xr (cid:12)(cid:12) n − − (cid:17) dx − + Z B cγ u | x | dx ! − (26)Moreover, for function u β ( x ) defined in Proposition 4.1 for m = 2 − n , inequality (26)becomes equality.Let us mention that Hardy inequality (26) has the same leading term in the right handside as in inequality (3), but inequality (26) is sharp one.Finally, it is difficult to compare inequality (4) with (26), but (26) is sharp one, i.e.,for the functions u β ( x ) defined in Proposition 4.1, inequality (26) becomes an equality. Acknowledgement
This paper is partially supported by the National Scientific Program”Information and Communication Technologies for a Single Digital Market in Science, Educationand Security (ICTinSES)”, contract No DO1205/23.11.2018, financed by the Ministry of Educationand Science in Bulgaria and also by the Grant No BG05M2OP001–1.001–0003, financed by theScience and Education for Smart Growth Operational Program (2014-2020) in Bulgaria and co-financed by the European Union through the European Structural and Investment Funds.
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