A note on Hardy s inequalities with boundary singularities

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1 A note on Hardy s inequalities with boundary singularities Mouhamed Moustaha Fall Abstract. Let be a smooth bounded domain in R N with N 1. In this aer we study the Hardy-Poincaré inequalities with weight function singular at the boundary of. In articular we give sufficient conditions so that the best constant is achieved. Key Words: Hardy inequality, extremals, -Lalacian. 1 Introduction Let be a domain in R N, N 1, with and > 1 a real number. In this note, we are interested in finding minima to the following quotient u dx λ u dx 1.1) µ λ, ) := inf u W 1, ) x u dx, in terms of λ R and. If λ =, we have the -Hardy constant u dx 1.2) µ, ) = inf u W 1, ) x u dx Université Catholique de Louvain-La-Neuve, déartement de mathématique. Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgique. mouhamed.fall@uclouvain.be, mouhamed.m.fall@gmail.com. 1

2 which is the best constant in the Hardy inequality for mas suorted by. The existence of extremals for µ λ,2 ) was studied in [1] while for µ,2 ), one can see for instance [6], [5], [21] and [19] for µ,n ). Given a unit vector ν of R N, we consider the half-sace H := {x R N : x ν }. For N = 1, the following Hardy inequality is well known 1.3) ) 1 t u dt u dt u W 1,, ). Moreover µ, H) = 1 for historical comments also. ) is the H-Hardy constant and it is not achieved, see [15] For N 2, it was recently roved by Nazarov [2] that the H-Hardy constant is not achieved and 1.4) µ, H) := inf V W 1, S N 1 + ) S N 1 + N ) 2 V 2 + σ V 2 ) 2 dσ S N 1 + V dσ where S+ N 1 is an N 1)-dimensional hemishere. Notice that this roblem always has a minimizer by the comact embedding L S N 1 1, + ) W S N 1 + ). The quantity µ, H) is exlicitly known only in some secial cases. Indeed, µ,2 H) = N 2 4 while for = N then µ N,N H) is the first Dirichlet eigenvalue of the oerator div u N 2 u) in W 1,N ) with the standard metric. S N 1 + Problem 1.1) carries some similarities with the questions studied by Brezis and Marcus in [2], where the weight is the inverse-square of the distance from the boundary of and = 2. We also deal with this roblem in the resent aer for all > 1 in Aendix A. We generalize here the existence result obtained by R.Musina and the author in [1] for any > 1 and N 1. Theorem 1.1 Let > 1 and be a smooth bounded domain in R N, N 1, with. There exits λ, ) [, + ) such that 1.5) µ λ, ) < µ, H), λ > λ, ). The infinimum in 1.1) is attained for any λ > λ, )., 2

3 The existence of λ, ) comes from the fact that su µ λ, ) = µ, H), λ R see Lemma 2.2. Now observe that the maing λ µ λ, is non-increasing. Moreover, for bounded domains, letting λ 1 be the first Dirichlet eigenvalue of the - Lalace oerator div u 2 u) in W 1, ), it is lain that µ λ1,) =. Then we define λ, ) := inf{λ R : µ λ, ) < µ, H)} so that µ λ, < µ, H) for all λ > λ, ). In articular λ, ) λ 1. On the other hand there are various bounded smooth domains with such that λ, ) [, ), see Proosition ) 2.5 and Proosition 2.6. Furthermore if N = 1 then µ, R \ {}) = = µ, H) thus λ, ). 1 It is obvious that if is contained in a half-ball centered at the origin then µ, ) = µ, H) thus λ, ) and in addition u dx µ, H) λ, ) = inf u W 1, ) u dx We have obtained the following result. x u dx. Theorem 1.2 If is contained in a half-ball centered at the origin then there exists a constant cn, ) > such that 1.6) λ, ) cn, ) diam). The constant cn, ) aearing in 1.6) has the roerty that cn, 2) is the first Dirichlet eigenvalue of in the unit disc of R 2. This tye of estimates was first roved by Brezis-Vàzquez in [3] when = 2, N 2 and later on, extended to the case 1 < < N by Gazzola-Grunau-Mitidieri in [13] when dealing with µ, R N \{}) := N. More recisely they roved the existence of a ositive constant CN, ) such that for any oen subset of R N, there holds 1.7) u µ, R N \ {} ) x u CN, ) 3 ) ωn N u u W 1, ),

4 where is the measure of and ω N the measure of the unit ball of R N. The constant CN, ) was exlicitly given and CN, 2) = cn, 2) as was obtained in [3]. The main ingredients to rove 1.7) is the Schwarz symmetrization and a dimension reduction via the transformation x u N ω, where ωx) = x satisfies div ω 2 ω) + µ, R N \ {} ) x ω 1 = in R N \ {}. For = 2, the lower bound in 1.6) was obtained in [1] by a similar transformation and using the Poincaré inequality on S N 1 +. However, in view of 1.4), such argument do not aly here when 2 and N. By analogy, to reduce the dimension, we will consider the maing x u N ) v, where vx) := x V x x is a weak solution to the equation div v 2 v) + µ, H) x v 2 v = in D H) whenever V is a minimizer of 1.4). Then exloiting the strict convexity of the maing a a, estimate 1.6), for 2, follows immediately while the case 1, 2) carries further difficulties as it can be seen in Section 2.2. The argument to rove the attainability of µ λ, ) is taken from de Valeriola- Willem [7]. It allows to show that, u to a subsequence, the gradient of the Palais- Smale sequences converges oint-wise almost every where. Therefore an alication of the Brezis-Lieb lemma with some simles arguments yields the existence of extremals. 2 Hardy inequality with one oint singularity Let C be a roer cone in R N, N 2 and ut Σ := C S N 1. It was shown in [2] that the C-Hardy constant is not achieved and it is given by 2.1) µ, C) = inf V W 1, Σ) Σ N ) 2 V 2 + σ V 2 ) 2 dσ Σ V dσ. 4

5 Letting V W 1, Σ) be the ositive minimizer to this quotient then the function ) 2.2) vx) := x N x V x satisfies 2.3) v 2 v h = µ, C) C C x v 1 h h C 1 c C). Notice that µ,2 C) = ) N 2 2+λ1 2 Σ), where λ 1 Σ) is the first Dirichlet eigenfunction of the Lalace oerator on Σ endowed with the standard metric on S N 1. This was obtained in [21], [19] and [1]. 2.1 Existence In this Section we show that the condition µ λ, ) < µ, H) is sufficient to guaranty the existence of a minimizer for µ λ, ). We emhasize that throughout this section, can to be taken to be an oen set satisfying the uniform shere condition at. Namely there are balls B + and B R N \ such that B + B = {}. This holds if is of class C 2 at, see [[16] 14.6 Aendix]. We start with the following aroximate local Hardy inequality. Lemma 2.1 Let be a smooth domain in R N, N 1, with and let > 1. Then for any ε > there exits r ε > such that 2.4) µ, B rε )) µ, H) ε, where B r ) is a ball of radius r centered at. Proof. If N = 1 then 2.4) is an immediate consequence of 1.3). From now on we can assume that N 2. We denote by N the unit normal vector-field on. U to a rotation, we can assume that N ) = E N, so that the tangent lane of at coincides with R N 1 = san{e 1,..., E N 1 }. Denote by B + r = {y B r ) : y N > }. For r > small, we introduce the following system of coordinates centered at see [9]) via the maing F : B + r given by F y) = Ex ỹ) + y N N Ex ỹ)), 5

6 where ỹ = y 1,..., y N 1 ) and ỹ Ex ỹ) is the exonential maing of endowed with the metric induced by R N. This coordinates induces a metric on R N given by g ij y) = i F y), j F y) for i, j = 1,..., N. Let u Cc F B r + )) and ut vy) = uf y)) then 2.5) F B + r ) u dx = B + r v g g dy, F B + r ) x u dx = B + r F y) v g dy, with g stands for the determinant of the g while v g = g v, v) 2. Since F y) = y + O y 2 ) and g ij y) = δ ij + O y ), we infer that v g g dy v dy B r + B r F y) v 1 Cr) +, g dy y v dy B + r for some constant C > deending only on and. Furthermore since µ, B + r ) µ, H), using 2.5) we conclude that B + r µ, F B + r )) 1 Cr)µ, H). We are in osition to rove 1.5) in the following Lemma 2.2 Let be a smooth domain in R N, N 1, with and let > 1. Then there exists λ, ) [, + ) such that Proof. We first show that µ λ, ) < µ, H) λ > λ, ). 2.6) su µ λ, ) = µ, H). λ R Ste 1: We claim that su λ R µ λ, ) µ, H). For r > small, we let ψ C B r )) with ψ 1, ψ in R N \ B r ) and 2 6

7 ψ 1 in B r ). For a fixed ε > small, there holds 4 x u = x ψu + 1 ψ)u 1 + ε) x ψu + cε) 1 + ε) x ψu + cε) Now by 2.4) and hence 2.7) µ, H) ε) x 1 ψ) u u. µ, H) ε) x ψu ψu) x u 1 + ε) ψu) + cε) u. Since ψu) ψ u + u ψ ) we deduce that ψu) 1 + ε)ψ u + c u ψ 1 + ε) u + c u. Using 2.7), we conclude that 2.8) µ, H) ε) x u 1 + ε) 2 u + cε) u. This imlies that µ, H) su λ R µ λ, ) and the claim follows. Ste 2: We claim that su λ R µ λ, ) µ, H). Denote by ν the unit interior normal of. For δ we consider the cone C δ + := { x R N x ν > δ x } and ut Σ δ = C+ δ S N 1. For every η >, let V Cc Σ ) such that N ) ) 2 2 V 2 + σ V 2 dσ Σ µ, H) + η. V dσ Σ On the other hand, there exists δ > small such that su V Σ δ. From this we conclude that 2.9) µ, H) µ, C δ +) µ, H) + η. 7

8 Since is smooth at, for every δ >, there exists r δ > such that C δ + B r ) for all r, r δ ). Clearly by scale invariance, µ, C+ B δ r )) = µ, C+). δ For ε >, we let φ W 1, C+ δ B r )) such that φ dx C+ δ Br) µ, C+) δ + ε. x φ dx From this we deduce that Since C δ + B r) µ λ, ) C δ + B r) C δ + B r) x φ dx r µ C δ +) + ε + λ φ dx λ C δ + B r) φ dx x φ dx C+ δ Br) φ dx C δ + B r) C δ + B r). x φ dx C+ δ Br) φ dx, we get µ λ, ) µ, C δ +) + ε + r λ. The claim follows immediately by 2.9). Therefore 2.6) is roved. Finally as the ma λ µ λ, ) is non increasing while µ λ1,) = < µ, H), we can set λ, ) := inf{λ R : µ λ, ) < µ, H)} so that λ, ) < µ, H) for any λ > λ, ). Remark 2.3 Observe that the roof of Lemma 2.2 highlights that lim µ, B r )) = µ, H) = lim µ λ,). r λ Proof of Theorem 1.1 Let λ > λ, ) so that µ λ, ) < µ, H). We define the maings F, G : 8

9 W 1, ) R by and F u) = Gu) = u λ u x u. By Ekeland variational rincial, there is a minimizing sequence u n W 1, ) normalized so that and with the roerties that Gu n ) = 1, n N F u n ) µ λ, ), 2.1) Ju n ) = F u n ) µ λ, )G u n ) in W 1, )). U to a subsequence, we can assume that there exists u W 1, ) such that 2.11) u n u in L ), u n u in L ) and u n u a.e. in. Moreover by 2.8), we may assume that x 1 u n x 1 u in L ). We set θ n = u n u and { s if s 1 T s) = s s if s > 1. It follows that for every r ) T θ n ) r. Moreover notice that un 2 u n u 2 u ) T θ n ) = Ju n ), T θ n ) + µ λ, ) + λ Therefore by 2.1), 2.11) and 2.12) we infer that un 2 u n u 2 u ) T θ n ). 9 x u 2 n u n T θ n ) u n 2 u n T θ n ) u 2 u T θ n ).

10 Consequently by [7]-Theorem 1.1, 2.13) lim u n n By Brezis-Lieb Lemma [4] 2.14) 1 lim n ) θ n = x θ n = u. x u. Fix ε > small. By 2.8) and Rellich, there exists λ ε such that µ, H) ε) x θ n θ n λ ε θ n = θ n + o1). Using this together with 2.13) and 2.14) we get µ λ, ) x u u λ u u n θ n λ u n + o1) F u n ) µ, H) ε) x θ n + o1) ) µ λ, ) µ, H) ε) 1 x u + o1) µ λ, ) µ, H) + ε + µ, H) ε) x u + o1). Send n and then ε to get µ λ, ) µ, H)) x u µ λ, ) µ, H). Hence x u 1 because µ λ, ) µ, H) < and the roof is comlete. As a consequence of the existence theorem, we have Corollary 2.4 Let be a smooth bounded domain of R N, N 2, with. Then µ, R N \ {} ) = N < µ, ) µ, H). Proof. By 2.6) < µ, ) µ, H). If the strict inequality holds, then there exists a ositive minimizer u W 1, ) for µ, ) by Theorem 1.1. But then µ, R N \ {} ) < µ, ), because otherwise a null extension of u outside would achieve the Hardy constant in R N \ {} which is not ossible. 1

11 As mentioned earlier, we shall show that there are smooth bounded domains in R N such that λ, ) [, ). These domains might be taken to be convex or even flat at. For that we let ν S N 1 and δ, r, R >. We consider the sector 2.15) Cr,R δ := { x R N x ν > δ x, r < x < R }. Proosition 2.5 Let N 2 and > 1. Then for all δ, 1), there exist r, R > such that if a domain contains Cr,R δ then µ,) < µ, H). Proof. Consider the cone C δ := { x R N x ν > δ x } Notice that by Harnack inequality µc δ ) < µc δ ) for any δ < δ < 1. Thus for any δ, 1), we can find u Cc C δ ) such that u C δ < µ, H). x u C δ Hence we choose r, R > so that su u Cr,R δ. By Corollary 2.4, starting from exterior domains, one can also build various examle of ossibly annular) domains for which λ, ) <. The following argument is taken in [Ghoussoub-Kang [14] Proosition 2.4]. If U R N, N 2, is a smooth exterior domain the comlement of a smooth bounded domain) with U then by scale invariance µ, U) = µ, R N \{}). We let B r ) a ball of radius r centered at the and define r := B r ) U then clearly the ma r µ r ) is decreasing with 2.16) µ, R N \ {} ) = inf µ, r ) and µ, H) = su µ, r ). r> We have the following result for which the roof is similar to the one given in [14] by Corollary 2.4 and Harnack inequality. Proosition 2.6 There exists r > such that the maing r µ, r ) is leftcontinuous and strictly decreasing on r, + ). In articular µ, R N \ {}) < µ, r ) < µ, H), r r, + ). r> 11

12 2.2 Remainder term We know that for domains contained in a half-ball λ, ). Our aim in this section is to obtain ositive lower bound for λ, ) by roviding a remainder term for Hard s inequality in these domains. In [13], Gazzola-Grunau-Mitidieri roved the following imroved Hardy inequality for 1 < < N: 2.17) u µ, R N \ {} ) ) x u ωn N CN, ) u, that holds for any bonded domain of R N and u W 1, ). Here the constant CN, ) > is exlicitly given while CN, 2) is the first Dirichlet eigenvalue of of the unit disc in R 2. We shall show that such tye of inequality holds in the case where the singularity is laced at the boundary ) of the domain. To this end, we will use the function vx) := x N V defined in 2.2) to reduce the dimension. x x Throughout this section, we assume that N 2 since the case N = 1 was already roved by Tibodolm [22] Theorem 1.1. Indeed, he showed that 1 u r) dr µ, H) 1 We start with conic domains r ur) dr 1)2 1 = {x = rσ R N r, 1), σ Σ }, ur) dr, u W 1,, 1). where Σ is a domain roerly contained in S N 1 and having a Lischitz boundary. ) We will denote by V the ositive minimizer of 2.1) in Σ while vx) := x N V satisfies 2.3) in the infinite cone {x = rσ R N r, + ), σ Σ }. Finally we remember that by Harnack inequality 1 v L loc ). x x Recall the following inequalities see [17] Lemma 4.2) which will be useful in the remaining of the aer. Let [2, ) then for any a, b R N 2.18) a + b a + If 1, 2) then for any a, b R N b + a 2 a b. 2.19) a + b a b 2 + c) a + b ) 2 + a 2 a b. We first make the following observation. 12

13 Lemma 2.7 Let u Cc ), u. Set ψ = u v then If 2 2.2) u µ, ) x u v ψ, 1 If 1 < < ) u µ, ) x u v ψ c) CΣ 2 v ψ + ψ v ) 2, Proof. We rove only the case 2 as the case 1, 2) goes similarly. Notice that u = v ψ +ψ v then we use the inequality 2.18) with a = v ψ and b = ψ v to get u ψ v + ψ v 2 1 ψ v v ψ) v ψ. 1 It is lain that ψ v 2 ψ v v ψ) = v 2 v v ψ ) = v 2 v vψ ) ψ v. Inserting this in the first inequality and using 2.3) we deduce that u v ψ + v 2 v vψ ) 1 C Σ v ψ + µ, ) x u. 1 The imrovement in the case 2 is an immediate consequence of the above lemma. Lemma 2.8 For all 2 u µ, ) x u where Λ := inf f C 1 c,1) 1 r 1 f dr 1 r 1 f dr. Λ 2 1 u, u Cc ), 1 13

14 Proof. Since u u, we may assume that u. We only need to estimate the right hand side in 2.2). We use olar coordinates x x, x x ) = r, σ) and denote by r the radial direction. Then using 2.18), v ψ = Σ Σ Λ 1 r 1 V ψ r r + σ ψ 1 V r 1 ψ r Λ Σ 1 The lemma readily follows from 2.2). Σ u r N 1 = Λ u. 1 V r 1 ψ It is easy to see that by integration by arts Λ 1 while for integer N then Λ corresonds to the first Dirichlet eigenvalue of in the unit ball of R. We now turn to the case 1, 2) which carries more difficulties. We shall need the following intermediate result. Lemma 2.9 Let 1, 2) and u Cc ), u. Setting ψ = u v a constant c = c, Σ) > such that c r ψ v r 2 )/2 v ψ. then there exists Proof. Let ψ := r 1 ψ and use ψ v as a test function in the weak equation 2.3). Then by Hölder ψ v µ, ) r v ψ + ψ v 1 v ψ ) 1 ) 1 µ, ) r v ψ + ψ v v ψ. Therefore by Young s inequality, for ε > small there exists a constant C ε > deending on and Σ such that 1 εc)) ψ v C ε r v ψ + C ε v ψ. 14

15 Recall that ψ = r 1 ψ. Then since ψ c) r 1 ψ + r ψ ), we conclude that there exists a constant c = c, Σ) such that 2.22) c r ψ v r 1 v ψ + r 2 )/ v ψ, we have used the fact that r r 2 )/ for all r, 1). To estimate the first term in the right hand side in 2.22) we will use the 2-dimensional Hardy inequality. Through the olar coordinates x r, σ) r 1 v ψ = = Σ 1 ) ψ V r 1 r r 1 ) ψ V r Σ r 1 2 V ψ r r Σ 1 2 V v v ψ r = 2 Σ 1 Σ r N +1 v ψ. To conclude, we notice that r N +1 = r N 2 r 2 )/ r N 1 r 2 )/ as 1, 2) so that r 1 v ψ 2 r 2 )/ v ψ. Inserting this in 2.22) the lemma follows immediately. We are now in osition to rove the imroved Hardy inequality for 1, 2). Lemma 2.1 Let 1, 2). Then there exists a constant c = c, Σ) > such that u µ, ) x u c u, u Cc ). Proof. Here also we may assume that u. We need to estimate the right hand 15

16 side of 2.21). Let r = x then by Hölder and Lemma 2.9, we have r 2 2 v ψ = v ψ CΣ 2 2 r 2 v ψ + ψ v ) 2 )/2 2 )/2 v ψ + ψ v ) v ψ CΣ 2 ) /2 v ψ + ψ v ) 2 r v ψ + ψ v v ψ CΣ 2 ) /2 ) 2 )/2 v ψ + ψ v ) 2 ) 2 )/2 2 1 r v ψ r ψ v v ψ c CΣ 2 ) /2 2 )/2 v ψ + ψ v ) 2 r 2 2 v ψ ), where c a ositive constant deending only on and Σ and we have used once more the fact that r r 2 )/ for all r, 1). Consequently by 2.21), we deduce that 2.23) u µ, ) x u c r 2 2 v ψ. To roceed we estimate 1 Σ u r N 1 = 1 1 V r 1 ψ c) V r ψ r Σ Σ 1 c) V r 2 ψr Σ c) r 2 2 v ψ. The first inequality comes from the 2-dimensional embedding W 1, L 2 2 L 3, one can see [[13] age 2155] for the roof. Putting this in 2.23) we conclude that there exists a ositive constant c = c, Σ) such that u µ, ) x u c u which was the urose of the lemma. 16

17 The main result in this section is contained in the next theorem. Theorem 2.11 Let be a domain in R N with. If is contained in a half-ball centered at then there exists a constant cn, ) > such that u µ, H) x u cn, ) diam) u u W 1, ). Proof. Let R = diam) be the diameter of. Then is contained in a half ball B + R that of radius R centered at the origin. From Lemma 2.8 and Lemma 2.1 we infer B + R u µ, H) x u cn, ) B + R u u C R B + c ) R by homogeneity. The theorem readily follows by density. We do not know whether diam) might be relaced with ω N 1 N as in [13] at least when is convex and 2. There might exists also logarithmic imrovement as was recently obtained in [11] inside cones and = 2. One can see also the work of Barbatis-Filias-Tertikas in [1] for domains containing the origin or when x is relaced by the distance to the boundary. A Hardy s inequality We denote by d the distance function of : dx) := inf{ x σ : σ }. In this section, we study the roblem of finding minima to the following quotient u dx λ u dx A.1) ν λ, ) := inf u W 1, ) d u dx, where > 1 and λ R is a varying arameter. Existence of extremals to this roblem was studied in [2] when = 2 and in [18] with λ =. It is known see for instance [18]) that ν, ) c for any smooth bounded domain while for ) convex domain, the Hardy constant ν, ) is not achieved and ν, ) = =: c. The main result in this section is contained in the following 17 1

18 Theorem A.1 Let be a smooth bounded domain in R N λ, ) [, + ) such that and > 1, there exits ) 1 A.2) ν λ, ) <, λ > λ, ). The infinimum in A.1) is attained if λ > λ, ). We start with the following result which is stronger than needed. It was roved in [2] for = 2 and in [12] when 2 < N as the authors were dealing with Hardy-Sobolev inequalities. Lemma A.2 Let be a smooth bounded domain in R N and 1, ). Then there exists β = β, ) > small such that A.3) u c β d u β where β := {x : dx) < β}. u H 1 ), Proof. Since u u, we may assume that u. Let u Cc ) and ut v = d 1 u. Using 2.18) and 2.19), we get A.4) u c d u c)d 1 v + 1 A.5) u c d u d v 2 c) ) c v + d v 1 1 d v ) if 2, d v ) if 1, 2). By integration by arts, we have d v ) = β d v + β v c β v + β v, β for a ositive constant deending only on. Multily the identity divd d) = 1 + d d by v in integrate by arts to get 1 + o1)) v = d v 1 d v+ d v c) d v 1 v + d v. β β β β β 18

19 By Hölder and Young s inequalities A.6) 1 + o1) cε) v c ε β d v + β d v. β Case 2. Using A.6) we infer that 1 + o1) cε) v c ε β β d 1 v + β β v. β It follows from A.4) that for ε, β > small u c β d u c β d 1 v + β v β ) as desired. Case 1, 2). By Hölder and Young s inequalities β d v = β d v 1 c v + d v d 2 v 2 c ε 1 β c v + d v ) 2 ) 2 1 c v + d v ) 2 + εc ) 2 ) 2 v + εc d v β β and thus 1 cε) d v c ε β d β 2 v 2 1 c v + d v ) 2 + εc β v. Using this in A.6) we obtain 1 + o1) cε) v d v cβ β β 2 ) cβ v. c v + d v β By A.5), we conclude that for ε, β > small u c d u d v c β β β 2 ) c v. c v + d v β This ends the roof of the lemma. 19

20 Lemma A.3 Let be a bounded domain of class C 2 in R N. λ, ) [, + ) such that Then there exists ν λ, ) < c λ > λ, ). Proof. The roof will be carried out in 2 stes. Ste 1: We claim that su λ R ν λ, ) c. For β > we define β := {x : dx) < β}. Let ψ C β ) with ψ 1, ψ in R N \ β 2 small, there holds d u = d ψu + 1 ψ)u 1 + ε) d ψu + C 1 + ε) d ψu + C By A.3), we infer that and hence A.7) c c d ψu ψu) and ψ 1 in β. For ε > 4 d 1 ψ) u u. d u 1 + ε) ψu) + C u. Since ψu) ψ u + u ψ ) we deduce that ψu) 1 + ε)ψ u + C u ψ 1 + ε) u + C u. Using A.7), we conclude that c d u 1 + ε) 2 This means that c su λ R ν λ, ). Ste 2: We claim that su λ R ν λ, ) c. u + Cε, β) u. 2

21 Let β > then by 1.3) and scale invariance we have µ,, β) = c. Hence for ε > there exits a function φ W 1,, β) such that A.8) c + ε β φ ds β s φ ds. Letting ux) = φdx)), there exists a ositive constant C deending only on such that Furthermore β u = β d u = β β By A.8) we conclude that s φ s) dσ s 1 + Cβ) β s s φs) dσ s 1 Cβ) φ s) ds. β φs) ds. ν λ, ) u dx λ u dx β β d u dx β c + ε) 1 + Cβ u dx 1 Cβ + λ β. d u dx β Since d u 2 dx β β u dx, we get sending β to we get the desired result. ν λ, ) c + ε) 1 + Cβ 1 Cβ + β λ, Clearly the roof of Theorem A.1 goes similarly as the one of Theorem 1.1 and we ski it. It was shown in [2] that λ2, ) R and that ν λ, ) is not achieved for any λ λ2, ). On the other hand by [8], there are domains for which λ2, ) <, see also [18]. We oint out that if is convex then by [22] there exists a constant an, ) > exlicitly given) such that an, ) λ, ). N 21

22 We finish this section by showing that there are smooth bounded domains in R N such that λ, ) [, ). We let U R N, N 2 with U be an exterior domain and set r = B r ) U. Proosition A.4 Assume that > N+1 2 then there exists r > such that ν, r ) < ). 1 Proof. Clearly µ, R N \ {}) = N < ) such that 1 > N + ε so by 2.16), there exits r > such that µ, r ) < N ) 1 rovided > N+1 2. Let ε > ) 1 + ε <. The conclusion readily follows since ν, r ) µ, r ) because r. References [1] Barbatis G., Filias S., Tertikas A., A unified aroach to imroved L Hardy inequalities with best constants. Trans. Amer. Math. Soc., 356, 24), [2] Brezis H. and Marcus M., Hardy s inequalities revisited. Dedicated to Ennio De Giorgi. Ann. Scuola Norm. Su. Pisa Cl. Sci. 4) ), no. 1-2, [3] Brezis H. and Vàzquez J. L., Blow-u solutions of some nonlinear ellitic roblems. Rev. Mat. Univ. Comlut. Madrid ), no. 2, [4] Brezis H. and Lieb E., A relation between ointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc ), [5] Caldiroli P., Musina R., On a class of 2-dimensional singular ellitic roblems. Proc. Roy. Soc. Edinburgh Sect. A ), [6] Caldiroli P., Musina R., Stationary states for a two-dimensional singular Schrödinger equation. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 8) 4-B 21), [7] de Valeriola S. and Willem M., On Some Quasilinear Critical Problems, Advanced Nonlinear Studies 9 29),

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