On the uniform Poincaré inequality
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1 On the uniform Poincaré inequality Abdesslam oulkhemair, Abdelkrim Chakib To cite this version: Abdesslam oulkhemair, Abdelkrim Chakib. On the uniform Poincaré inequality. Communications in Partial Differential Equations, Taylor Francis, 2007, ), pp <hal > HAL Id: hal Submitted on 9 May 2006 HAL is a multi-disciplinary open access arive for the deposit and dissemination of scientific resear documents, whether they are published or not. The documents may come from teaing and resear institutions in France or abroad, or from public or private resear centers. L arive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau reere, publiés ou non, émanant des établissements d enseignement et de reere français ou étrangers, des laboratoires publics ou privés.
2 On the uniform Poincaré inequality A. oulkhemair 1 and A. Chakib 2 1 Laboratoire de Mathématiques Jean Leray, CNRS UMR6629/ Université de Nantes, 2, rue de la Houssinière, P 92208, Nantes, France. boulkhem@math.univ-nantes.fr 2 Département de Mathématiques Appliquées et Informatique FST de eni-mellal, Université Cadi-Ayyad.P. 523, eni-mellal, Morocco. akib@fstbm.ac.ma Abstract We give a proof of the Poincaré inequality in W 1,p ) with a constant that is independent of U, where U is a set of uniformly bounded and uniformly Lipsitz domains in R n. As a byproduct, we obtain the following : The first non vanishing eigenvalues λ 2 ) of the standard Neumann variational) boundary value problem on for the Laplace operator are bounded below by a positive constant if the domains vary and remain uniformly bounded and uniformly Lipsitz regular. 1 Introduction As it is well known, the Poincaré inequality is an important tool in the study of many problems of partial differential equations and numerical analysis. Others call it Poincaré-Friedris inequality. Recall that in the case of the usual Sobolev space H 1 ) = W 1,2 ), it says the following: There exists a positive constant C) su that u L 2 ) C) u L 2 ) for all u H 1 ) su that u dx = 0 or u = 0 on some reasonable subset of. 1) Here is say a regular connected open set in R n. In this paper, we discuss the problem of the dependence of the constant C) on without, however, looking for the best constant. In fact, in many applications, one needs to move the domain and often needs to apply Poincaré inequality with a constant independent of the varying domains. This commonly happens, for example, in free boundary problems or shape optimization problems. See for example [6, 7]. This also can happen when one simply needs to approximate the domain by more regular or more suitable ) ones. See 1
3 for example [8] where the authors studied a Robin problem on a Lipsitz domain and needed to apply a Poincaré or Friedris) inequality whose constant is uniform with respect to the sequence of regular domains whi approximates the Lipsitz one. There are essentially two methods for proving the Poincaré inequality. The first one, in fact the direct one, uses mainly the Taylor formula or Newton formula) via the density of the regular functions in the Sobolev space. When one can perform it, it leads to an explicit constant C). For example, for a general domain and under the condition u = 0 on, one obtains as C) an explicit function of the diameterof or its measure. One can also obtain an explicit C) when is convex and the condition u dx = 0 is used. However, for a general geometry and under this null integral condition, it seems to be difficult to find an explicit reasonable constant C). To get an idea, see the recent attempt [12] where a more or less general geometry of is used. The other method of proof is an argument by contradiction using functional analysis. See for example [4]. The advantage of this method is that it allows to treat the case of a general domain with a minimal regularity and under any condition like that in 1). However, there is a trouble with it: It gives no information on the way the constant C) depends on, and so, when moves we do not know what C) does in the general case. For a while, we have been following the point of view that consists in seeking an explicit constant C). We did not succeed. In this work, we solve our problem by considering a reasonable class of domains U for whi we prove the Poincaré inequality with a constant independent of U. Surprisingly, the method of proof is just the usual functional analysis argument by contradiction, in addition to few elementary teniques from shape optimization. The class U we consider is any set of domains that are uniformly bounded and have a uniform Lipsitz regularity. Thanks are due to G. Carron and R. Souam for discussions on the geometry related to this subject. 2 Notations and definitions We consider a fixed, open, bounded and regular subset of R n, whi will contain all our domains. y a domain, we shall always mean an open connected subset of R n. Let us recall some definitions and notations whi are needed below. Let ξ, y be vectors in R n su that ξ = 1 and ε be a strictly positive number. The set defined by Cy, ξ, ε) = {x R n ; x y)ξ x y cos ε and x y < ε} 2) is called the cone with vertex y, direction ξ and angle to the vertex and height ε. A domain is said to have the ε-cone property if for all x, there exists a direction ξ and a positive real number ε su that Cy, ξ, ε), for all y x, ε) 3) where x, ε) is the open ball of center x and radius ε in R n. Let us now recall some notions of convergence of sets. 2
4 A sequence E k ) of measurable subsets of R n is said to converge to a measurable subset E of R n in the sense of aracteristic functions, and this is denoted by E k where χ A denotes the aracteristic function of the set A. E, if χ Ek χ E in L 1 R n ), Let K 1 and K 2 be two compact subsets of R n, consider the usual distance from x R n to K i i = 1, 2) and set dx, K 1 ) = inf y K 2 dx, y), ρk 1, K 2 ) = sup x K 1 dx, K 2 ). dx, K 2 ) = inf x K 1 dx, y) y definition, the Hausdorff distance from K 1 to K 2, is the following non negative number d H K 1, K 2 ) = maxρk 1, K 2 ), ρk 2, K 1 )). It is well known that this indeed defines a distance between compact subsets of R n. Now, if K j, j N, and K are compact subsets of R n H, we shall write K j K if dh K j, K) 0 as j. Finally, if k ) k is a sequence of open subsets of and an open subset of, we say that the H sequence k ) k converges to in the Hausdorff complement) sense and we denote it by c k, if \ k 3 Uniform Poincaré inequality Let us define the class U by H \. U = { / is open, connected and satisfies the ε cone property}. Note that the elements of U are uniformly Lipsitz regular domains see for example [2, 3, 7, 11]). We shall need the following two lemmas. Lemma 1 If k ) k is a sequence in U, then, there exists a subsequence of k ) k, denoted again k ) k, and an element of U su that k H c, k and k H. Proof : Let k ) k be a sequence in U. It follows from a well known result in functional analysis related to shape optimization see for example [7, 10, 11]), that there exists a subsequence denoted again k ) k and an open subset of having the ε-cone property su that k H c, k and k H. According to Theorem 3.18 of [5], is connected. Since the domain is Lipsitz regular, it is also connected. 3
5 Lemma 2 If E k, k N, and E are measurable subsets of, E k E and f k ) k is sequence of functions whi converge to a function f in L 1 ), then we can extract subsequences denoted again E k ) k and f k ) k su that lim f k dx = f dx k E k E Proof : Since E k ) k is a convergent sequence to E in the sense of aracteristic functions, we can extract a subsequence denoted again E k ) k su that χ Ek χ E almost everywhere. Now, we have f k dx f dx = χ Ek f k χ E f dx E k E χ Ek χ E )f dx + χ Ek f k f) dx χ Ek χ E )f dx + f k f) dx. The lemma follows from the hypothesis and the Lebesgue convergence theorem. Let us denote by M the set of measurable subsets of. The main result of this paper is the following : Theorem 1 Let E : U M be a map whi is continuous in the sense that, if k, then, E k ) E). Assume also that E) and E) > 0, for all U. Then, there exists a constant C > 0, su that u 1 u dx C E) i u p,, u W 1,p ), U, 4) p, E) where p, denotes the norm in L p ), p 1, and A is the measure of the measurable set A. Proof : Assume that the result is false. Then, there exist sequences k ) k in U, and u k ) in W 1,p k ), su that, for all k 1, u 1 k u k dx > k E k ) E k ) i u k p,k. 5) p, 1 If v k ) k is the sequence defined by v k = u k u k dx, we have E k ) E k ) v k p,k > k i v k p,k, k 1. If w k ) k is the sequence defined by w k = v k v k p,k, we have, for all k 1, w k p,k = 1, 6) and i w k p,k < 1 k. 7) 4
6 Since the elements of U are uniformly Lipsitz regular, according to Chenais [3], we can construct a uniform extension operator P from W 1,p ) to W 1,p ), for all U, su that there exists a constant M > 0 independent of su that P u W 1,p ) M u W 1,p ). Set w k = P k w k. For all k 1, we have w k W 1,p ) M w k W 1,p ), ) M w k p, + i w k p,k M ), k hence, w k W 1,p ) 2 M, k 1. Since the sequence w k ) k is bounded in W 1,p ), there exists a subsequence denoted again w k ) k and an element w W 1,p ) w V ) when p = 1) su that w k w in L p ). In particular, i w k is convergent to i w in D ), for all i 1 i n), where D ) is the space of distributions in. Now, according to Lemma 1, the sequence k ) k contains a subsequence denoted again k ) k whi converges to U in the Hausdorff sense. Let us show that i w = 0 in, for all i, 1 i n. Indeed, let ϕ D) the space of C functions with compact support in ); it follows from the Hausdorff convergence of k ) k to, that there exists k 0 N su that ϕ D k ), k k 0. For all i, 1 i n, we have then i w, ϕ = i w i w k, ϕ + i w k, ϕ i w i w k, ϕ + i w k p,k ϕ p,, k k 0, where p is su that 1 p + 1 p = 1. Using 7) and passing to the limit, we obtain that i w, ϕ = 0, ϕ D) and i, 1 i n. Thus, w = 0 in. Since is connected, it means that w is constant in. Now, we can assume that k as it follows from Lemma 1. Therefore, applying the continuity of E and passing to the limit, we obtain that E) up to a set of Lebesgue measure 0. Let us show that w dx = 0. Since w k w in L p ) and E k ) E), it follows from Lemma 2 that E) E) w dx = lim k E k ) w k dx = 0. Since w is constant in and E) has a positive measure, we obtain that w = 0 in. Finally, it follows from 6) and Lemma 2 that 0 = w p dx = lim w k p dx = 1, k k 5
7 whi is a contradiction. This proves the theorem. We state as a corollary, the following particular cases of the uniform Poincaré inequality of Theorem 1. Corollary 1 i) Under the assumptions of Theorem 1, there exists a constant C > 0, su that u p, C i u p,, U and u W 1,p ), su that u = 0 in E). 8) ii) If E is a measurable subset of of positive measure, then, there exists a constant C > 0 su that n ) u p, C i u p, + E 1 p 1 u dx, u W 1,p ), U, E. 9) E iii) There exists a constant C > 0 su that n ) u p, C i u p, + 1 p 1 u dx, u W 1,p ), U. 10) Note that parts ii) and iii) of Corollary 1 correspond respectively to the cases E) = E and E) = of Theorem 1. Another consequence of Theorem 1 concerns the first non vanishing eigenvalue λ 2 ) of the standard Neumann variational) boundary value problem on for the Laplace operator. In fact, in the case p = 2 and E) =, it is well known that the best constant in the Poincaré inequality u 1 2 u dx C 2, i u 2 2, is the inverse of λ 2 ), see Dautray-Lions [4], pp , for example. So, in this case, we can paraphrase Theorem 1 as follows : Corollary 2 The eigenvalues λ 2 ) are bounded below by a positive constant if the domains vary and remain uniformly bounded and uniformly Lipsitz regular. Comment : Su a result may be of interest in geometry. Indeed, geometers usually seek lower bounds for the first eigenvalues of the standard Dirilet and Neumann problems. See, for example, [13], [14]. The lower bound given by Corollary 2 may seem to be rough in comparison with what is known in geometry. Of course, this is because the geometric and regularity assumption, that is the cone assumption, made on the domains is very weak. It is a difficult problem to determine the constant as a function of under su a general assumption. Next, we try to go further and extend Theorem 1 to some smaller subsets E) of. In fact, for simplicity, we state the result only for E) constant and equal to some hypersurface. One can certainly extend it to more general subsets by using the notion of capacity. 6
8 Theorem 2 Let be a Lipsitz hypersurface in R n, possibly with a boundary, su that and 0 < <, = dσ). Then, there exists a constant C > 0, su that u 1 u dσ C p, Proof : i u p,, u W 1,p ), p > 1, U, su that. 11) It follows the same seme as that of Theorem 1, so we shall be brief. Assuming the statement false, there exist sequences k ) in U su that k, k 1, and w k ) in W 1,p k ) su that, for all k 1, w k p,k = 1, i w k p,k < 1 k and w k dσ = 0. As before, we define the sequence w k ) whi is bounded in W 1,p ) and we can assume it to converge weakly to w W 1,p ). Also, by a similar argument, one can show that w is constant in, the limit of k for the Hausdorff convergence. Now, it follows from the compactness of the trace operator from W 1,p ) to L p ) and the facts that < and, that w dσ = w k dσ = 0; so that w = 0 in. Of course, this is in contradiction with the fact that lim k w p dx = lim w k p dx = 1. k k An immediate consequence of this theorem is the following Corollary 3 Under the hypothesis of Theorem 2, there exists a constant C > 0, su that i) u p, C n ) i u p, + 1 p 1 u dx, u W 1,p ), U,. 12) ii) u p, C i u p, U, and u W 1,p ), u = 0 on. 13) 4 Remarks 1. Let P : W 1,p ) W 1,p ) be a uniform extension operator su as that of Chenais. Then, the uniform Poincaré inequality u p, C 1 n i u p,, U, and u W 1,p ), u = 0 on, is equivalent to the following homogeneous estimate on the operator P : i P u p, C 2 n i u p,, U, and u W 1,p ), u = 0 on. 7
9 In fact, this relies essentially on the Poincaré inequality applied to the fixed domain. This remark may be of interest mainly in the domain of shape optimization. another possible way of proof of the uniform Poincaré inequality. It also gives 2. The above results also hold when p =, that is, in the case of Lipsitz functions. In fact, the proofs are rather elementary and one obtains an explicit constant C). For example, we can write ux) 1 E) E) uy) dy 1 E) E) ux) uy) dy ux) uy) diam) sup diam) u x y x y L ), for all u W 1, ). One can even replace the average of u on E) by the value of u on some point x, since the functions are continuous. References [1] Adams, R.A., Sobolev spaces, 65, Academic Press, New York, [2] Chenais, D., On the Existence of a Solution in a Domain Identification Problem, J. Mat. Annal. Appl., 52, 2) ). [3] Chenais, D., Sur une famille de variétés à bord lipsitziennes. Application à un problème d identification de domaine, Ann. Institut Fourier, 27, 4) ). [4] Dautray, R. and Lions, J.L., Analyse mathématique et calcul numérique pour les sciences et les teniques, Masson, Paris, [5] Falconer, K. K., The geometry of fractal sets, Cambridge university press, [6] Haslinger, J. and Neittaanmaki, P., Finite Element Approximtion for Optimal Shape Design. Theory and Applications, John and sons LTD, [7] Henrot, A. and Pierre, M., Variation et optimisation de formes-une analyse géométrique, Springer Series: Mathématiques et Applications, 48, 2005 [8] Lanzani, L. and Shen, Z., On the Robin boundary condition for Laplace s equation in Lipsitz domains, Comm. Partial Differential Equations, ) ). [9] Neças, J., Les méthodes directes en théorie des équations elliptiques, Masson, Paris, [10] Ly, I. and Seck, D., Isoperimetric inequality for an interior free boundary problem with p- Laplacian operator, Electron. J. Differential Equations 109) 12 pp. 2004). [11] Pironneau, O., Optimal Shape Design for Elliptic Systems, springer series in computational physics, springer-verlag, [12] Weiying, Z. and He, Q. On Friedris-Poincaré-type inequalities, Journal of Mathematical Analysis and Applications, 304 2) ). [13] Gallot, S., Minorations sur le λ 1 des variétés riemanniennes, Séminaire ourbaki, 33e année, 1980/81, no 569. [14] Meyer, D., Minoration de la première valeur propre non nulle du problème de Neumann sur les variétés riemanniennes à bord, Annales de l institut Fourier, 36, no 2, 1986), p
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