AAnisotropic Hardy inequalities
Francesco Della Pietra ∗ ,Giuseppina di Blasio † ,Nunzia Gavitone ‡ A bstract . We study some Hardy-type inequalitiesinvolving a general norm in R n and an anisotropicdistance function to the boundary. The case of theoptimality of the constants is also addressed. MSC: D , D Keywords: Hardy inequalities, anisotropic operators.
Let F be a smooth norm of R n . In this paper we investigate thevalidity of Hardy-type inequalities (cid:90) Ω F ( ∇ u ) dx (cid:62) C F ( Ω ) (cid:90) Ω u d dx , ∀ u ∈ H ( Ω ) , ( . )where Ω is a domain of R n , and d F is the anisotropic distance tothe boundary with respect the dual norm (see Section for the ∗ Università degli studi di Napoli Federico II, Dipartimento di Matematicae Applicazioni “R. Caccioppoli”, Via Cintia, Monte S. Angelo -
Napoli,Italia. Email: [email protected] † Seconda Università degli studi di Napoli, Dipartimento di Matema-tica e Fisica, Via Vivaldi, - Caserta, Italia. Email: giusep-pina.diblasio@unina .it ‡ Università degli studi di Napoli Federico II, Dipartimento di Matematicae Applicazioni “R. Caccioppoli”, Via Cintia, Monte S. Angelo -
Napoli,Italia. Email: [email protected] a r X i v : . [ m a t h . A P ] D ec introduction precise assumptions and definitions). We aim to study the bestpossible constant for which ( . ) holds, in the sense that C F ( Ω ) = inf u ∈ H ( Ω ) (cid:90) Ω F ( ∇ u ) dx (cid:90) Ω u d dx .In the case of the Euclidean norm, that is when F = E = | · | ,the inequality ( . ) reduces to (cid:90) Ω | ∇ u | dx (cid:62) C E ( Ω ) (cid:90) Ω u d E dx , ∀ u ∈ H ( Ω ) , ( . )where d E ( x ) = inf y ∈ ∂Ω | x − y | , x ∈ Ω is the usual distance function from the boundary of Ω , and C E is the best possible constant.The inequality ( . ) has been studied by many authors, un-der several points of view. For example, it is known that forany bounded domain with Lipschitz boundary Ω of R n , E ( Ω ) (cid:54) (see [D , MMP , BM ]). In particular, if Ω is aconvex domain of R n , the optimal constant C E in ( . ) is inde-pendent of Ω , and its value is C E = , but there are smoothbounded domains such that C E ( Ω ) < (see [MMP , MS ]).Furthermore, in [MMP ] it is proved that C E is achieved if andonly if it is strictly smaller than .Actually, the value of the best constant C E ( Ω ) is still for amore general class of domains. This has been shown, for exam-ple, in [BFT ], under the assumption that d E is superharmonicin Ω , in the sense that ∆d E (cid:54) in D (cid:48) ( Ω ) . ( . )As proved in [LLL ], when Ω is a C domain the condition ( . )is equivalent to require that ∂Ω is weakly mean convex, that isits mean curvature is nonnegative at any point.The fact that the constant C E ( Ω ) = is not achieved haslead to the interest of studying “improved” versions of ( . ), byadding remainder terms which depend, in general, on suitablenorms of u . For instance, if Ω is a bounded and convex set, in[BM ] it has been showed that (cid:90) Ω | ∇ u | dx (cid:62) (cid:90) Ω u d E dx + (cid:90) Ω u dx , ∀ u ∈ H ( Ω ) ,( . ) introduction where L is the diameter of Ω . Actually, in [HHL ] the authorsshowed that the value can be replaced by a constant whichdepends on the volume of Ω , namely c ( n ) | Ω | − ; here c ( n ) issuitable constant depending only on the dimension of the space(see also [FMT ]).When Ω satisfies condition ( . ), several improved versions of( . ) can be found for instance in [BFT , BFT , FMT ]. Moreprecisely, in [BFT ] the authors proved that (cid:90) Ω | ∇ u | dx − (cid:90) Ω | u | d E dx (cid:62) (cid:90) Ω | u | d E (cid:18) log D d E (cid:19) − dx , ( . )where D (cid:62) e · sup { d E ( x , ∂Ω ) } and u ∈ H ( Ω ) .The aim of this paper is to study Hardy inequalities of the type( . ), and to show improved versions in the anisotropic settinggiven by means of the norm F , in the spirit of ( . ). For example,one of our main result states that for suitable domains Ω of R n ,and for every function u ∈ H ( Ω ) , it holds that (cid:90) Ω F ( ∇ u ) dx − (cid:90) Ω u d dx (cid:62) (cid:90) Ω u d (cid:18) log Dd F (cid:19) − dx , ( . )where D = e · sup { d F ( x , ∂Ω ) , x ∈ Ω } .The condition we will impose on Ω in order to have ( . ) willinvolve the sign of an anisotropic Laplacian of d F (see sections and ). We will also show that such condition is, in general, notequivalent to ( . ).Actually, we will deal also with the optimality of the involvedconstants. Moreover, we will show that ( . ) implies an im-proved version of ( . ) in terms of the L norm of u , in thespirit of ( . ). More precisely, we will show that if Ω is a convexbounded open set then (cid:90) Ω F ( ∇ u ) dx − (cid:90) Ω | u | d dx (cid:62) C ( n ) | Ω | − (cid:90) Ω | u | dx .We emphasize that Hardy type inequalities in anisotropic set-tings have been studied, for example, in [V , DPG , BF ],where, instead of considering the weight d − , it is taken into ac-count a function of the distance from a point of the domain (seefor example [BFT , GP , BV , HLP, VZ , AFV , BCT ]and the reference therein for the Euclidean case).The structure of the paper is the following. In Section wefix the necessary notation and provide some preliminary results notation and preliminaries which will be needed later. Moreover, we discuss in some de-tails the condition we impose on Ω in order to be ( . ) and ( . )true. In Section we study the inequality ( . ) and we give someapplications. In Section the improved versions of ( . ) are in-vestigated and, finally, Section is devoted to the study of theoptimality of the constants in ( . ). Throughout the paper we will consider a convex even -homogeneousfunction ξ ∈ R n (cid:55)→ F ( ξ ) ∈ [ , + ∞ [ ,that is a convex function such that F ( tξ ) = | t | F ( ξ ) , t ∈ R , ξ ∈ R n , ( . )and such that α | ξ | (cid:54) F ( ξ ) , ξ ∈ R n , ( . )for some constant . Under this hypothesis it is easy to seethat there exists α (cid:62) α such that H ( ξ ) (cid:54) α | ξ | , ξ ∈ R n .Furthermore we suppose that F is strongly convex, in the sensethat F ∈ C ( R n \ { } ) and ∇ F > 0 in R n \ { } .In this context, an important role is played by the polar func-tion of F , namely the function F o defined as x ∈ R n (cid:55)→ F o ( x ) = sup ξ (cid:54) = ξ · xF ( ξ ) .It is not difficult to verify that F o is a convex, -homogeneousfunction that satisfies | ξ | (cid:54) F o ( ξ ) (cid:54) | ξ | , ∀ ξ ∈ R n . ( . )Moreover, the hypotheses on F ensures that F o ∈ C ( R n \ { } ) (see for instance [S ]) F ( x ) = ( F o ) o ( x ) = sup ξ (cid:54) = ξ · xF o ( ξ ) . notation and preliminaries The following well-known properties hold true: F ξ ( ξ ) · ξ = F ( ξ ) , ξ (cid:54) = , ( . ) F ξ ( tξ ) = sign t · F ξ ( ξ ) , ξ (cid:54) = , t (cid:54) = , ( . ) ∇ F ( tξ ) = | t | ∇ F ( ξ ) ξ (cid:54) = , t (cid:54) = , ( . ) F (cid:0) F oξ ( ξ ) (cid:1) = , ∀ ξ (cid:54) = , ( . ) F o ( ξ ) F ξ (cid:0) F oξ ( ξ ) (cid:1) = ξ ∀ ξ (cid:54) = . ( . )Analogous properties hold interchanging the roles of F and F o .The open set W = { ξ ∈ R n : F o ( ξ ) < 1 } is the so-called Wulff shape centered at the origin. More gener-ally, we denote W r ( x ) = r W + x = { x ∈ R : F o ( x − x ) < r } ,and W r ( ) = W r .We recall the definition and some properties of anisotropic cur-vature for a smooth set. For further details we refer the reader,for example, to [ATW ] and [BP ]. Definition . . Let A ⊂ R n be a open set with smooth boundary.The anisotropic outer normal n A is defined as n A ( x ) = ∇ ξ F ( ν A ( x )) , x ∈ ∂A ,where ν A is the unit Euclidean outer normal to ∂A . Remark . . We stress that if A = W r ( x ) , by the properties of F it follows that n A ( x ) = ∇ ξ F (cid:0) ∇ ξ F o ( x − x ) (cid:1) = ( x − x ) , x ∈ ∂A .Finally, let us recall the definition of Finsler Laplacian ∆ F u = div ( F ( ∇ u ) F ξ ( ∇ u )) , ( . )defined for u ∈ H ( Ω ) . Definition . . Let u ∈ H ( Ω ) , we say that u is ∆ F -superharmonicif − ∆ F u (cid:62) in D (cid:48) , ( C H )that is (cid:90) Ω F ( ∇ u ) F ξ ( ∇ u ) · ∇ ϕ dx (cid:62) , ∀ ϕ ∈ C ∞ ( Ω ) , ϕ (cid:62) .Similarly, by writing that u is ∆ -superharmonic we will meanthat u is superharmonic in the usual sense, that is − ∆u (cid:62) . notation and preliminaries Anisotropic distance function
Due to the nature of the problem, it seems to be natural to con-sider a suitable notion of distance to the boundary.Let us consider a domain Ω , that is a connected open set of R n , with non-empty boundary.The anisotropic distance of x ∈ Ω to the boundary of ∂Ω isthe function d F ( x ) = inf y ∈ ∂Ω F o ( x − y ) , x ∈ Ω . ( . )We stress that when F = | · | then d F = d E , the Euclideandistance function from the boundary.It is not difficult to prove that d F is a uniform Lipschitz func-tion in Ω and, using the property ( . ), F ( ∇ d F ( x )) = a.e. in Ω . ( . )Obviously, assuming sup Ω d F < + ∞ , d F ∈ W , ∞ ( Ω ) and thequantity r F = sup { d F ( x ) , x ∈ Ω } , ( . )is called anisotropic inradius of Ω .For further properties of the anisotropic distance function werefer the reader to [CM ].The main assumption in this paper will be that d F is ∆ F -super-harmonic in the sense of Definition . . Remark . . We emphasize that if Ω is a convex set, the func-tions d E and d F are respectively ∆ and ∆ F -superharmonic. Thiscan be easily proved by using the concavity of d E and d F in Ω (see for instance [EG ], and [DPG ] for the anisotropic case).Actually, in the Euclidean case there exist non-convex sets forwhich d E is still ∆ -superharmonic. An example can be obtainedfor instance in dimension n = , taking the standard torus (see[AK ]). Similarly, in the following example we show that thereexists a non-convex set such that the anisotropic distance func-tion d F is ∆ F -superharmonic. Example . . Let us consider the following Finsler norm in R F ( x , x , x ) = ( x + x + a x ) ,with a > 0 ; then F o ( x , x , x ) = (cid:18) x + x + x a (cid:19) . notation and preliminaries We consider the set Ω ⊂ R obtained by rotating the ellipse γ = { ( , x , x ) : ( x − R ) + x a < r } with R > r > 0 ,about the x − axis. Obviously Ω is not convex. In order to showthat ( C H ) holds, we first observe that if we fix a generic point x ∈ γ , then being F isotropic with respect to the first two componentsthe anisotropic distance is achieved in a point x of the boundaryof γ . Moreover it is not difficult to show that the vector x − x hasthe same direction of the anisotropic normal n Ω (see Definition . , and see also [DPG ]). Hence by Remark . d F ( x ) = r − F o ( x − x ) ,where x = ( , R , ) is the center of the ellipse.Now let us introduce a plane polar coordinates ( ρ , ϑ ) such thatto a generic point Q = ( x , x , x ) ∈ R , it is associated the point Q (cid:48) = ( ρ cos ϑ , ρ sin ϑ , x ) , where ρ = (cid:113) x + x and ϑ ∈ [ , ] .Then, by construction, Ω = { Q (cid:48) ∈ R : F o ( Q (cid:48) − C ) < r } ,where C = ( R cos ϑ , R sin ϑ , ) and F o ( Q (cid:48) − C ) = ( R − ρ ) + x a .Then as observed before, fixed Q (cid:48) ∈ Ωd F ( Q (cid:48) ) = r − F o ( Q (cid:48) − C ) == r − (cid:115) ( R − ρ ) + x a = r − (cid:115)(cid:18) R − (cid:113) x + x (cid:19) + x a .Now we are in position to prove ( C H ). We note that for all Q (cid:48) (cid:54) = C ∆ F d F ( Q (cid:48) ) = div ( F ( ∇ d F ) F ξ ( ∇ d F ))= ∂ d F ∂x + ∂ d F ∂x + a ∂ d F ∂x =
1ρ ∂d F ∂ρ + ∂ d F ∂ρ + a ∂ d F ∂x = R − o ( Q (cid:48) − C ) . ( . )Being ρ > R − r , we get that d F is ∆ F -superharmonic in Ω if R > 2r for all a > 0 . anisotropic hardy inequality Remark . . In general, if Ω is not convex, to require that d F is ∆ F -superharmonic does not assure that d E is ∆ -superharmonic.Indeed, let Ω be as in Example . ; if we take R (cid:62) then, asshowed before, − ∆ F d F (cid:62) . On the other hand it is possible tochoose a > 0 such that d E is not ∆ -superharmonic. To do that,it is enough to prove that the mean curvature of Ω is negative atsome points of the boundary for a suitable choice of a . Indeedin [LLL ] it is proved that d E is ∆ -superharmonic on Ω if andonly if the mean curvature H Ω ( y ) (cid:62) for all y ∈ ∂Ω .The parametric equations of ∂Ω are ϕ ( t , ϑ ) : x = ( R + r cos ϑ ) cos t = φ ( ϑ ) cos tx = ( R + r cos ϑ ) sin t = φ ( ϑ ) sin tx = ar sin ϑ = ψ ( ϑ ) ,where t , ϑ ∈ [ , ] .Then for y = ϕ ( t , ϑ ) ∈ ∂Ω we have H Ω ( y ) = − φ ( φ (cid:48)(cid:48) ψ (cid:48) − φ (cid:48) ψ (cid:48)(cid:48) ) − ψ (cid:48) (( φ (cid:48) ) + ( ψ (cid:48) ) ) | φ | (( φ (cid:48) ) + ( ψ (cid:48) ) ) = ar ( R + cos ϑ + r cos ϑ ( a − )) | R + r cos ϑ | ( r sin ϑ + a r cos ϑ ) . ( . )Finally we observe that if ϑ = π then H Ω ( y ) < 0 , if a > 1 . Theorem . . Let Ω be a domain in R n and suppose that condi-tion ( C H ) holds. Then for every function u ∈ H ( Ω ) the followinganisotropic Hardy inequality holds (cid:90) Ω F ( ∇ u ) dx (cid:62) (cid:90) Ω u d dx ( . ) where d F is the anisotropic distance function from the boundary of Ω defined in ( . ) .Proof. First we prove inequality ( . ). Being F convex, we havethat F ( ξ ) (cid:62) F ( ξ ) + ( ξ ) F ξ ( ξ ) · ( ξ − ξ ) . anisotropic hardy inequality Hence putting ξ = ∇ u and ξ = Au ∇ d F d F , with A positive con-stant, recalling that F ( ∇ d F ) = , by the homogeneity of F weget (cid:90) Ω F ( ∇ u ) dx (cid:62) − A (cid:90) Ω u d dx + A (cid:90) Ω F F ξ ( ∇ d F ) · ∇ u dx .By the Divergence Theorem (in a general setting, contained forexample in [A ]) we have (cid:90) Ω F ( ∇ u ) dx (cid:62) − A (cid:90) Ω u d dx + A (cid:90) Ω F ξ ( ∇ d F ) d F · ∇ ( u ) dx (cid:62) − A (cid:90) Ω u d dx − A (cid:90) Ω u ∆ F d F d F dx + A (cid:90) Ω u d dx Being − ∆ F d F (cid:62) we get (cid:90) Ω F ( ∇ u ) dx (cid:62) ( A − A ) (cid:90) Ω u d dx .Then maximizing with respect to A we obtain that A = , andthen ( . ) follows. Remark . . We observe that if Ω is a convex domain in R n , aninequality of the type ( . ) can be immediately obtained by usingthe following classical Hardy inequality involving the Euclideandistance function d E (cid:90) Ω | ∇ u | dx (cid:62) (cid:90) Ω u d E dx . ( . )By ( . ) we easily get (cid:90) Ω F ( ∇ u ) dx (cid:62) α (cid:90) Ω | ∇ u | dx (cid:62) α (cid:90) Ω u d E dx (cid:62)
14 α α (cid:90) Ω u d ,( . )where the constant in the right-hand side is smaller than since α < α . We emphasize that if Ω is not convex inequality ( . )holds under the assumption that d E is E -superharmonic, since( . ) is in force. On the other hand the assumption on d E is notrelated with the hypotesis required about d F in the Theorem . ,as observed in Remark . .Using Theorem . it is not difficult to obtain a lower boundfor the first eigenvalue of ∆ F defined in ( . ). hardy inequality with a reminder term Corollary . . Let Ω be a bounded domain of R n and suppose thatcondition ( C H ) holds. Let λ ( Ω ) be the first Dirichlet eigenvalue ofthe Finsler Laplacian, that is λ ( Ω ) = min u ∈ H ( Ω ) u (cid:54) = (cid:90) Ω [ F ( ∇ u )] dx (cid:90) Ω | u | dx . ( . ) Then λ ( Ω ) (cid:62) , where r F is the anisotropic inradius of Ω defined in ( . ) .Proof. Let v the first eigenfunction related to λ ( Ω ) such that (cid:107) v (cid:107) L ( Ω ) = . Then ( . ) and inequality ( . ) imply λ ( Ω ) = (cid:90) Ω [ F ( ∇ v )] dx (cid:62) (cid:90) Ω v d dx (cid:62) ,that is the claim. Theorem . . Let Ω be a domain of R n . Let us suppose also thatcondition ( C H ) holds, and sup { d F ( x , ∂Ω ) , x ∈ Ω } < + ∞ . Then forevery function u ∈ H ( Ω ) the following inequality holds: (cid:90) Ω F ( ∇ u ) dx − (cid:90) Ω u d dx (cid:62) (cid:90) Ω u d (cid:18) log Dd F (cid:19) − dx , ( . ) where D = e · sup { d F ( x , ∂Ω ) , x ∈ Ω } .Proof. We will use the following notation: X ( t ) = − log t , t ∈ ] , [ .Being F convex, we have that F ( ξ ) (cid:62) F ( ξ ) + ( ξ ) F ξ ( ξ ) · ( ξ − ξ ) .Let us consider ξ = ∇ u , ξ = u2 ∇ d F d F (cid:20) − X (cid:18) d F D (cid:19)(cid:21) . hardy inequality with a reminder term Being d F ( x ) (cid:54) De , by the -homogeneity of F we get F ( ∇ u ) dx (cid:62)
14 u d F ( ∇ d ) (cid:20) − X (cid:18) d F D (cid:19)(cid:21) + ud F (cid:20) − X (cid:18) d F D (cid:19)(cid:21) ×× F ( ∇ d ) F ξ ( ∇ d ) · (cid:18) ∇ u − u2 ∇ d F d F (cid:20) − X (cid:18) d F D (cid:19)(cid:21)(cid:19) == −
14 u d (cid:20) − X (cid:18) d F D (cid:19)(cid:21) ++ ud F (cid:20) − X (cid:18) d F D (cid:19)(cid:21) F ξ ( ∇ d F ) · ∇ u ( . )where last equality follows by F ( ∇ d F ) = , the -homogeneity of F and property ( . ). Let us observe that, using the DivergenceTheorem (in a general setting, contained for example in [A ]),we have (cid:90) Ω ud F (cid:20) − X (cid:18) d F D (cid:19)(cid:21) F ξ ( ∇ d F ) · ∇ u dx == − (cid:90) Ω u div (cid:18)(cid:20) − X (cid:18) d F D (cid:19)(cid:21) F ξ ( ∇ d F ) d F (cid:19) dx == (cid:90) Ω u (cid:12) (cid:20) − X (cid:18) d F D (cid:19) + X (cid:18) d F D (cid:19)(cid:21) F ξ ( ∇ d F ) · ∇ d F d +− (cid:20) − X (cid:18) d F D (cid:19)(cid:21) ∆ F d F d F (cid:13) dx (cid:62)(cid:62) (cid:90) Ω
12 u d (cid:20) − X (cid:18) d F D (cid:19) + X (cid:18) d F D (cid:19)(cid:21) dx , ( . )where last inequality follows using the condition − ∆ F d F (cid:62) .Integrating ( . ), and using ( . ) we easily get (cid:90) Ω F ( ∇ u ) dx (cid:62) (cid:90) Ω
14 u d ×× (cid:14) − (cid:20) − X (cid:18) d F D (cid:19)(cid:21) + − (cid:18) d F D (cid:19) + (cid:18) d F D (cid:19) (cid:15) dx == (cid:90) Ω u d dx + (cid:90) Ω u d X (cid:18) d F D (cid:19) dx ,and the proof is completed. optimality of the constants Remark . . We observe that if Ω is a convex domain in R n ,arguing as in Remark . , an inequality of the type ( . ) can beimmediately obtained by using the following improved Hardyinequality involving d E contained in [BFT ] (cid:90) Ω | ∇ u | dx − (cid:90) Ω | u | d E dx (cid:62) (cid:90) Ω | u | d E (cid:18) log D d E (cid:19) − dx , ( . )where D (cid:62) e · sup d E ( x , ∂Ω ) and u ∈ H ( Ω ) . Obviously also inthis case it is not possible to obtain the optimal constants. Corollary . . Under the same assumptions of Theorem . , the fol-lowing anisotropic improved Hardy inequality holds (cid:90) Ω F ( ∇ u ) dx − (cid:90) Ω | u | d dx (cid:62) (cid:90) Ω | u | dx , ( . ) where r F is the anisotropic inradius defined in ( . ) .Proof. By Theorem . , to prove ( . ) it is sufficient to show that (cid:90) Ω | u | d (cid:18) log Dd F (cid:19) − dx (cid:62) (cid:90) Ω | u | dx .This is a consequence of the monotonicity of the following func-tion f ( t ) = − t log (cid:18) t e · r F (cid:19) , (cid:54) r F .Indeed f is strictly increasing and its maximum is r F . This con-cludes the proof.An immediate consequence of the previous result is containedin the following remark. Remark . . Let Ω ⊂ R n be a bounded convex domain. Thenthere exists a positive constant C ( n ) > 0 such that for any u ∈ H ( Ω ) we have (cid:90) Ω F ( ∇ u ) dx − (cid:90) Ω | u | d dx (cid:62) C ( n ) | Ω | − (cid:90) Ω | u | dx . Here we prove the optimality of the constants and of the expo-nent which appear in the Hardy inequality ( . ). More precisely,we prove the following result: optimality of the constants Theorem . . Let Ω be a piecewise C domain of R n . Suppose thatthe following Hardy inequality holds: (cid:90) Ω F ( ∇ u ) dx − A (cid:90) Ω u d dx (cid:62) B (cid:90) Ω u d (cid:18) log Dd F (cid:19) − γ dx , ∀ u ∈ H ( Ω ) , ( . ) for some constants A > 0 , B (cid:62) , γ > 0 , where D = e · sup { d F ( x , ∂Ω ) , x ∈ Ω } . Then: ( T ) A (cid:54) ; ( T ) If A = and B > 0 , then γ (cid:62) ; ( T ) If A = and γ = , then B (cid:54) .Proof. The proof is similar to the one obtained in the Euclideancase by [BFT ]. For the sake of completeness, we describe it indetails. As before, let us denote by X ( t ) = − log t , t ∈ ] , [ .In order to prove the results we will provide a local analysis.Hence, we fix a Wulff shape W δ of radius δ centered at a point x ∈ ∂Ω . Being ∂Ω piecewise smooth, we may suppose that for asufficiently small δ , ∂Ω ∩ W δ is C . Let now ϕ be a nonnegativecut-off function in C ∞ ( W δ ∩ Ω ) such that ϕ ( x ) = for x ∈ W δ/2 .First of all, we prove some technical estimates which will beuseful in the following. For ε > 0 and β ∈ R let us consider J β ( ε ) = (cid:90) Ω ϕ d − + X − β ( d F /D ) dx . ( . )We split the proof in several claims. Claim . The following estimates hold:(i) c ε − − β (cid:54) J β ( ε ) (cid:54) c ε − − β , for β > − , where c , c arepositive constants independent of ε ;(ii) J β ( ε ) = + β + ( ε ) + O ε ( ) , for β > − ;(iii) J β ( ε ) = O ε ( ) , for β < − . Proof of Claim . By the coarea formula, J β ( ε ) = (cid:90) δ0 r − + X − β ( r/D ) (cid:18) (cid:90) d F = r ϕ | ∇ d F | dH n − (cid:19) dr optimality of the constants Being F ( ∇ d F ) = , by ( . ), (cid:54) | ∇ d F | (cid:54) α and (cid:54) (cid:90) d F = r ϕ | ∇ d F | dH n − (cid:54) C .Then if β < − , (iii) easily follows. Moreover, if β > − , per-forming the change of variables r = Ds , (i) holds.As regards (ii), let us observe that ddr X β = β X β + r .Recalling that = F ( ∇ d F ) F ξ ( ∇ d F ) · ∇ d F , then ( β + ) J β ( ε ) = − (cid:90) Ω ϕ d F ( ∇ d F ) F ξ ( ∇ d F ) · ∇ [ X − β − ( d F /D )] dx == (cid:90) Ω div (cid:0) ϕ d F ( ∇ d F ) F ξ ( ∇ d F ) (cid:1) X − β − ( d F /D ) dx == (cid:90) Ω ϕd X − β − ( d F /D ) F ( ∇ d F ) F ξ ( ∇ d F ) · ∇ ϕdx ++ (cid:90) Ω ϕd − X − β − ( d F /D ) dx ++ (cid:90) Ω ϕ d X − β − ( d F /D ) ∆ F d F dx == O ε ( ) + β + ( ε ) .We explicitly observe that (cid:90) Ω ϕ d X − β − ( d F /D ) ∆ F d F dx = O ε ( ) being d F a C function in a neighborhood of the boundary (see[CM ]). Then (ii) holds.In the next claim we estimate the left-hand side of ( . ) when u = U ε , with U ε ( x ) = ϕ ( x ) w ε ( x ) , w ε ( x ) = d + εF X − θ ( d F ( x ) /D ) ,
12 < θ < 1 .Let us define Q [ U ε ] := (cid:90) Ω (cid:18) F ( ∇ U ε ) −
14 U d (cid:19) dx . Claim . The following estimates hold: Q [ U ε ] (cid:54) θ2 J − ( ε ) + O ε ( ) , as ε → ; ( . ) (cid:90) W δ ∩ Ω F ( ∇ U ε ) dx (cid:54)
14 J ( ε ) + O ε (cid:0) ε − (cid:1) , as ε → . ( . ) optimality of the constants Proof of Claim . The convexity of F implies that F ( ξ + η ) (cid:54) F ( ξ ) + ( ξ ) F ( η ) + F ( η ) , ∀ ξ , η ∈ R n .Hence by the homogeneity of F , (cid:90) Ω F ( ∇ U ε ) dx (cid:54)(cid:54) (cid:90) W δ ∩ Ω ϕ F ( ∇ w ε ) dx + (cid:90) W δ ∩ Ω w F ( ∇ ϕ ) dx ++ (cid:90) W δ ∩ Ω ε F ( ∇ ϕ ) F ( ∇ w ε ) dx == (cid:90) W δ ∩ Ω ϕ d − X − ( d F /D ) (cid:18) ε + − θX ( d F /D ) (cid:19) dx + I + I .As matter of fact, I (cid:54) C (cid:90) W δ ∩ Ω d X − ( d F /D ) dx = O ε ( ) ;similarly, also I = O ε ( ) . Then Q [ U ε ] (cid:54)(cid:54) (cid:90) W δ ∩ Ω ϕ d − X − ( d F /D ) (cid:34)(cid:18) ε + − θX ( d F /D ) (cid:19) − (cid:35) dx + O ε ( ) = (cid:54) (cid:90) W δ ∩ Ω ϕ d − X − ( d F /D ) ( ε − θX ( d F /D )) dx ++ (cid:90) W δ ∩ Ω ϕ d − X − ( d F /D ) ( ε − θX ( d F /D )) + O ε ( ) == a + a + O ε ( ) . ( . )Using (ii) of Claim with β = − + we get a = O ε ( ) ε → . ( . )As regards a , similarly, applying (ii) of Claim with β = − in the first time and β = − in the second time we obtain a = θ2 (cid:90) W δ ϕ d − X − ( d F /D ) dx + O ε ( ) . ( . )Then ( . ) follows by ( . ),( . ),( . ) and ( . ). Finally observingthat (cid:90) W δ ∩ Ω F ( ∇ U ε ) dx = Q [ U ε ] +
14 J ( ε ) , ( . ) eferences then the inequality ( . ) follows from ( . ) and (ii) of Claim .Now we are in position to conclude the proof of the Theorem.Since inequality ( . ) holds for any u ∈ H ( Ω ) we take as testfunction U ε . Then by ( . ) and (i) of Claim we have A (cid:54) ( ε ) (cid:90) W δ ∩ Ω F ( ∇ U ε ) dx (cid:54) + O ε ( ε ) .Letting ε → we obtain ( T ) .In order to prove ( T ) we put A = and reasoning by contra-diction we assume that γ < 2 . As before by ( . ) and (i) of Claim , we have (cid:54) Q [ U ε ] J − γ ( ε ) (cid:54) C ε − ε γ − − = Cε − γ → as ε → ,which is a contradiction and then γ (cid:62) .To conclude the proof of the Theorem we just have to prove ( T ) .If A = and γ = then by ( . ) we have B (cid:54) Q [ U ε ] J − ( ε ) (cid:54) θ2 J − ( ε ) + O ε ( ) J − ( ε ) .Then by assumption on θ and (i) of Claim , letting ε → weget B (cid:54) θ2 .Hence ( T ) follows by letting θ → . Remark . . We stress that Theorem . assures that the involvedconstants in ( . ) are optimal, and also in the anisotropic Hardyinequality ( . ). Actually, the presence of the remainder term in( . ) guarantees that the constant in ( . ) is not achieved. Acknowledgement.
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