HARDY INEQUALITIES WITH BOUNDARY TERMS. x 2 dx u 2 dx. (1.2) u 2 = u 2 dx.

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1 Electronic Journal of Differential Equations, Vol. 003(003), No. 3, pp ISSN: UL: or ftp ejde.math.swt.edu (login: ftp) HADY INEQUALITIES WITH BOUNDAY TEMS ZHI-QIANG WANG & MEIJUN ZHU Abstract. In this note, we present some Hardy type inequalities for functions which do not vanish on the boundary of a given domain. We establish these inequalities for both bounded and unbounded domains and also obtain the best embedding constants in these inequalities for special domains. Our results are motivated by and building upon some recent work in [5, 6, 9, 1]. 1. Introduction The standard Hardy inequality states that for N 3, (N ) N u x dx u dx (1.1) N for any u C0 ( N ). Here (N ) / is the best possible constant. This inequality can be extended to functions in the space D 1, ( N ) which is the completion of C0 ( N ) with respect to the norm u u dx. There are many generalizations of this inequality, see for example, [, 3,, 6, 7, 8, 10, 11] and references therein. The weighted version of this inequality was given in []. In this paper we will consider another type of generalizations (motivated by recent work of Li and Zhu [9] and Zhu [1]). Let N be a bounded domain and u D 1, 0 (). Since we can trivially extend u to a new function in D1, ( N ) which vanishes outside, we obtain the following Hardy inequality on a bounded domain: (N ) u x dx u dx. (1.) Naturally, one may ask whether there are some analogous inequalities that hold for function u H 1 () (Notice that u(x) may not vanish on the boundary of ). Since (1.) does not hold for any constant function, we shall expect, like in the case of Sobolev inequality (see, for example, [9]), the right hand side may include some lower order terms. 000 Mathematics Subject Classification. 35J0, 35J5. Key words and phrases. Hardy inequality with boundary terms, weighted versions of Hardy inequality. c 003 Southwest Texas State University. Submitted December, 00. Published April 16,

2 ZHI-QIANG WANG & MEIJUN ZHU EJDE 003/3 We shall consider a more general version of the Hardy inequality the weighted version ([]): for a < N, it holds for all u C 0 ( N ) (N a) x (a+1) u dx x a u dx. N N ecently, in [5, 6] a new formulation of this inequality has been given by using a conformal transformation. Based on this conformal transformation we first establish weighted Hardy type inequalities with boundary terms in two specific domains. Denote B 1 (0) {x N : x < 1}, and B1(0) c N \ B 1 (0). We assume below N when we treat the weighted version of the Hardy inequality and N 3 when we treat the classical Hardy inequality. Let us define the weighted Sobolev space Da 1, ( N ) to be the completion of C0 ( N ) with respect to the following norm u a x a u dx. In the following, for simplicity of notations we omit the integration variables when the situation is clear. Theorem 1.1. Let a < N. Then for all u D1, a ( N ) (N a) u x < x a u + N a (a+1) and (N a) B 1(0) Bc1(0) u B 1(0) x < x a u N a (a+1) B1 c(0) B 1(0) B 1(0) u, (1.3) u. (1.) emark 1.. The strict inequalities are due to the non-existence of extremal functions in (1.3) and (1.). The constants involved in the above inequalities are sharp in the sense that (N a) B inf x a 1(0) u + N a B u 1(0) u Da 1, ( N )\{0} and a similar statement holds for (1.). B 1(0), u x (a+1) Using similar arguments we obtain a Hardy inequality on any C 1 smooth domains with bounded boundary and 0 /. Theorem 1.3. Let a < N. If N is a smooth domain with being bounded and 0 /, then there is a constant C h (depending on ), such that for all u Da 1, ( N ) (N a) u x x a u + C (a+1) h u. (1.5) If in addition N is bounded and star-shaped with respect to the origin, then there is a constant C h > 0 (depending on ), such that for all u D1, a ( N ) (N a) c u x x a u C (a+1) h u. (1.6) c

3 EJDE 003/3 HADY INEQUALITIES WITH BOUNDAY TEMS 3 A natural question following the theorem is what may happen if is convex (thus star-shaped with respect to any interior point) but the origin lies outside? Based on a new integral inequality established in [1], we have the following result. Theorem 1.. Let N be a bounded piecewise smooth domain which contains the origin. Assume that consists of two smooth hyper-surfaces Γ 1 and Γ. If Γ is concave with respect to the domain and is part of the boundary of a rotationally symmetric convex domain, then /N (N ) holds for any u D 1, ( N ) with u 0 on Γ 1. u x u (1.7) emark 1.5. Let N be a smooth convex revolution solid which does not contain the origin. As a simple corollary of Theorem 1., we see that for any u D 1, ( N ), /N (N ) c u x u. (1.8) c For any domain and any u D 1, ( N ) \ {0}, let I(u, ) : u. u x We will give an example of a domain that satisfies the conditions in Theorem 1., (N ) inf I(u, ) <. (1.9) u D 1, ( N )\{0},u0 on Γ 1 Quite similar to the case of Sobolev inequality, we have the following theorem. Theorem 1.6. Let N be a smooth domain such that is bounded and 0 /. If 0 < inf u D 1, ( N )\{0} I(u, ) < (N ) /, then the infimum is achieved by a function ū D 1, ( N ). emark 1.7. Assume that satisfies the conditions in Theorem 1.. Following the proof of Theorem 1.6, we easily prove that if inf I(u, ) < (N ) /, u D 1, ( N )H 1 ()\{0}, u0 on Γ 1 then inf u D 1, ( N )\{0}, u0 on Γ 1 I(u, ) is achieved by some functions. This indicates that if N is a convex domain which does not contain the origin, then there might be no uniform lower bound for inf u D 1, ( N )\{0} I(u, c ).. Proofs of Theorems The proofs of Theorem are based on the following conformal transformation which was used in [5, 6] to give a new formulation of a family of weighted Sobolev inequalities due to Caffarelli-Kohn-Nirenberg in []. This family of inequalities include the weighted version of the Hardy inequalities. We define ϕ : N C : S N 1 as the conformal transformation x ϕ(x) ( ln x, ). (.1) x

4 ZHI-QIANG WANG & MEIJUN ZHU EJDE 003/3 Here we use (t, θ) S N 1. And we define u(x) x N a x v( ln x, x ), x N. (.) Due to the density lemma in [6, Lemma.1], we need to prove Theorem only for functions in C 1 0( N ). Proof of Theorem 1.1. Let u C 1 (B 1 (0)), and v be given by (.). By [6, Proposition.], we know that v H 1 ( ), where {(t, θ) S N 1 : t > 0}. Denote S : 0 S N 1, we have x a u ( θ v + (v t + N a v) )dµ B 1 ( v + (N )v t v + ( N a ) v )dµ and N a (v ) t dµ ( v + ( N a 0 S t ) v )dµ + N a (v ) t dµ, (.3) N a (v ) t dθdt N a v dθ, S where S t : t S N 1 and dµ dθdt. Also, it is easy to check that u dθ x N++a v dθ v dθ. B 1 B 1 S Therefore, we have x a u + N a B 1 On the other hand It follows that B 1 x a u dx + N a B 1 u x (a+1) dx B 1 u dθ u dx B 1 x (a+1) B 1 u dθ ( v + ( N a v dµ. ) v )dµ. ( v + ( N a ) v )dµ v dµ > ( N a ) which yields (1.3). The last inequality in the above expression follows from v being in H 1 ( ). The inequality (1.) can be proved in the same spirit, and we shall omit the details. Proof of Theorem 1.3. Without loss of generality, we can assume that B 1 (0). For any u(x) C 1 (), let ϕ be the transformation given by (.1), and v(x) be given by (.). Denote ϕ(). Thus. Similar to (.3), we have x a u ( θ v + (v t + N a v) )dµ ( v + ( N a ) v )dµ + N a (v ) t dµ.

5 EJDE 003/3 HADY INEQUALITIES WITH BOUNDAY TEMS 5 But due to Green s formula N a (v ) t dµ N a (v, 0) ηds ω, where η is the unit out norm vector of and ds ω is the volume element on. Then N a (v, 0) ηds ω N a v ds ω N a x (N a) u ( ln x, x/ x )ds ω N a x 1 a u N a C 0 u, where C 0 { (max{ x : x }) 1 a if 1 a 0, (min{ x : x }) 1 a if 1 a < 0. Therefore, x a u + N a C 0 u ( v + ( N a ) v )dµ. On the other hand, u dx v dµ. x (a+1) Above two inequalities yield (1.5). (1.6) can be proved in the same spirit, and we shall omit details here. Proof of Theorem 1.. We need to prove the inequality only for non-negative smooth functions. Suppose that u C 1 () is a non-negative function satisfying u 0 on Γ 1. Let be the ball centered at the origin which has the same volume as. Let u be the Schwartz symmetrization of u. Namely, we define u (x) sup{t : µ(t) > ω N x N }, where ω N is the volume of the unit ball in N, and µ(t) is the Lebesgue measure of the set {x : u(x) > t}. Then, it is well-known (see, e.g., Bandle [1]) that u x (u ) x. On the other hand, from Zhu [1] (this is the place where we use the assumption on Γ ) we know that u /N u. Since u 0 on, we know from the standard Hardy inequality that ( N ) (u ) x u. These three inequalities yield Theorem 1..

6 6 ZHI-QIANG WANG & MEIJUN ZHU EJDE 003/3 There might be a guess that for satisfying the condition in Theorem 1., inf I(u, ) u D 1, ( N )\{0},u0 on Γ 1 will always be larger than or equal to (N ) /. However, we present an example to show that this is not the case. An example. Let {(x, x N ) N : x N > 1}. We are going to show that u inf N 1 (N ). u D 1, ( N )\{0} < u x Let ϕ be the transformation given by (.1), and v(x) be given by (.) for x. Denote C 1 ϕ(). Then u C 1 ( v + ( N ) v )dµ + N C 1 (v ) t dµ. C 1 v dµ u x Now, we choose 0, t 0 (t )/ 0, 0 t ṽ(t, θ) 1, t ( + 0 t)/ 0, t + 0 0, t + 0, where 0 > /(N ), and will be chosen sufficiently large. A simple calculation shows that ( ṽ + ( N ) ṽ )dµ + N (ṽ ) t dµ C 1 C 1 ( N C 1 ) ṽ dµ + ( 3 + o(1)) SN 1 ( o(1))n S N 1, where o(1) 0 as. Let ũ(x) x N ṽ( ln x, x x ). It is easy to see that ũ D 1, ( N ). Thus for sufficiently large u ũ inf N 1 N 1 < ( N ). u D 1, ( N )\{0} ũ x u x When (supp ũ) o, where (supp ũ) o is the set of interior points of supp ũ, we easily see that inf I(u, ) < (N ) /. u D 1, ( N )\{0},u0 on Γ 1 Proof of Theorem 1.6. Let u m D 1, ( N ) be a minimizing sequence such that u m/ x 1. Then u m inf I(u, ) : ξ < ( N ). u D 1, ( N )\{0}

7 EJDE 003/3 HADY INEQUALITIES WITH BOUNDAY TEMS 7 If is bounded, u m C u m/ x C. Thus u m is uniformly bounded in H 1 (). It follows that u m ū weakly in H 1 (), thus ξ + o m (1) u m u m ū + ū + o m (1), (.) where o m (1) 0 as m. If is unbounded, since is bounded, contains the exterior of some ball domain. Then we may check that X {u u D 1, ( N )} is a Hilbert space with a norm u X u dx, this is due to the Sobolev inequality. Then u m is bounded in X and has a weak limit ū and we again have (.). By the weak convergence of um x ū x u m x to ū x in L (), we also have (u m ū) x + o m (1). (.5) Therefore, we have ξ + o m (1) u m ū + ū + o m (1) by (.) ( N ) u m ū x C h u m ū ū + ξ x + o m(1) (by Theorem 1.3 and the definition of ξ) ( N ) u m ū ū x + ξ x + o m(1) (by Sobolev embedding) [( N ) u m ū ξ] x + ξ + o m (1) (by (.5)), u m ū which implies x 0 as m. It follows that ū / x 1, thus ū is the minimizer of I(u, ). Notes Added in Proof. After this paper was accepted we found a paper by Adimurthi: Hardy-Sobolev inequality in H 1 () and its applications, Comm. Contem. Math., (00), 09-3, which contains related results and uses different methods. eferences [1] Bandle, C., Isoperimetric inequalities and applications, Pitman Monographs and studies in mathematics, V. 7, Pitman, Boston, [] Brezis, H., Marcus, M., Hardy s inequalities revisited. Dedicated to Ennio De Giorgi, Ann. Scuola Norm. Sup. Pisa Cl. Sci. () 5 (1997), [3] Brezis, H., Vázquez, J., Blow-up solutions of some nonlinear elliptic problems, ev. Mat. Univ. Complut. Madrid 10 (1997), [] Caffarelli, L., Kohn,. and Nirenberg. L, First order interpolation inequalities with weights, Compositio Math. 53(198), [5] Catrina, F. and Wang, Z.-Q., On the Caffarelli-Kohn-Nirenberg Inequalities, Comptes endus des Séances de l Académie des Sciences. Série I. Mathématique, 330(000), [6] Catrina, F. and Wang, Z.-Q., On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math. 5(001), [7] Chou, K. and Chu, C., On the best constant for a weighted Sobolev-Hardy inequality, J. London Math. Soc. (), 8(1993),

8 8 ZHI-QIANG WANG & MEIJUN ZHU EJDE 003/3 [8] Davies, E. B., A review of Hardy inequalities, The Mazýa anniversary collection, Vol. (ostock, 1998), 55 67, Oper. Theory Adv. Appl., 110, Birkhauser, Basel, [9] Li, Y. Y. and Zhu, M., Sharp Sobelev inequalities involving boundary terms, Geometric and Functional Analysis, 8(1998), [10] Lieb, E. and Yau, H-T., The stability and instability of relativistic matter, Comm. Math. Phys. 118(1988), no., [11] Lin, C. S., Interpolation inequalities with weights, Comm. Partial Diff. Equations, 11 (1986), [1] Zhu, M., Sharp Sobolev and isoperimetric inequalities with mixed boundary conditions, preprint. Zhi-Qiang Wang Department of Mathematics and Statistics, Utah State University, Logan, Utah , USA address: wang@math.usu.edu Meijun Zhu Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019, USA address: mzhu@math.ou.edu