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1 Max-Planck-Institut für Mathematik in den Naturwissenschaften Leipzig A Note on the Poincaré Inequality for Convex Domains by Mario Bebendorf Preprint no.: 24 23

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3 ANoteonthePoincaré Inequality for Convex Domains Mario Bebendorf March 13, 23 In this article a proof for the Poincaré inequality with explicit constant for convex domains is given. This proof is a modification of the original proof [5], which contains a mistake. AMS Subject Classification: 26D1 1 Introduction The classical proof for the Poincaré inequality u L 2 (Ω) c Ω u L 2 (Ω), where Ω R n is a bounded domain and u H 1 (Ω) with vanishing mean value over Ω, is based on the compact embedding of H 1 (Ω) in L 2 (Ω) which is valid under quite general assumptions on Ω (cf. [6]). However, the constant c Ω depends on the domain Ω, and the proof based on compactness does not provide insight into this dependency. For practical purposes it is important to know an explicit expression for this constant (see for example [2], [7]). Therefore, the special case of convex domains is interesting, since in [5] this constant is proved to be d/π, whered is the diameter of Ω. Though this proof is elegant, it contains a mistake. The same mistake can also be found in [1], in which the L 1 estimate is considered. The goal of this article is to fix the error made in [5]. Luckily, the constant d/π in the Poincaré inequality remains valid. Max Planck Institute for Mathematics in the Sciences, Inselstraße 22 26, 413 Leipzig, Germany (bebendorf@mis.mpg.de) 1

4 2 The one-dimensional case We first prove the Poincaré inequality for the one-dimensional case. In fact we will prove a generalization which the multidimensional case can be reduced to. Lemma 2.1 Let m N and ρ be a non-negative concave function on the interval [,L]. Then for all u H 1 (,L) satisfying it holds that ρ m (x)u(x)dx = (2.1) ρ m (x) u(x) 2 dx L2 ρ m (x) u (x) 2 dx. (2.2) π 2 Furthermore, the constant L 2 /π 2 is optimal. Proof. (a) Let us first assume that ρ is strictly positive and twice differentiable. Then each non-zero function v minimizing the quotient ρm (x) u (x) 2 dx ρm (x) u(x) 2 dx (2.3) and satisfying (2.1) must satisfy the Sturm-Liouville system (cf. [3]) [ρ m v ] + λρ m v = withv () = v (L) =, (2.4) where λ is the minimum of the quotient (2.3). After dividing (2.4) by ρ m and differentiating, we introduce the new variable w = ρ m/2 v and obtain w + m [ ρ (1+ 2 ρ m ) ] (ρ ) 2 w + λw = withw() = w(l) =. 2 ρ 2 Since ρ is concave, ρ. Hence, w + λw and integration by parts leads to λ w (x) 2 dx w(x) 2 dx. The last quotient is bounded by the first eigenvalue of the vibrating string with fixed ends, which gives λ π 2 /L 2. (b) If ρ is a non-negative concave function, we may represent it as the L -limit of strictly positive concave C 2 -functions ρ k, cf. [4]. From Part (a) one has ρ m k (x) û(x) 2 dx L2 ρ m π 2 k (x) u (x) 2 dx, 2

5 where û(x) :=u(x) u and u := ρm k (x)u(x)dx L ρm k (x)dx. Hence, ρ m k (x) u(x) 2 dx L2 ρ m π 2 k (x) u (x) 2 dx + u ρ m k (x)u(x)dx. In the limit k we obtain (2.2). (c) To see that the constant L 2 /π 2 is optimal, choose ρ m 1, L =1and u(x) =cos(πx). Then 1 ρm (x)u(x)dx =and 1 ρm (x) u(x) 2 dx 1 ρm (x) u (x) 2 dx = 1 1 cos2 (πx)dx π 2 1 sin2 (πx)dx = 1 π. 2 3 The n-dimensional case In the rest of this article we will consider the case n 2. By the following lemma we are able to reduce the n-dimensional problem to the onedimensional case. Lemma 3.1 Let Ω R n be a convex domain with diameter d. Assume that u L 1 (Ω) satisfies u(x)dx =. Then for any δ>there are disjoint Ω convex domains Ω i, i =1,...,k, such that Ω= k Ω i, i=1 Ω i u(x)dx =, i =1,...,k, and for each Ω i there is rectangular coordinate system such that Ω i {(x, y) R n : x d and y j δ, j =1,...,n 1}. Proof. For each α [, 2π] there is a unique hyperplane H α R n with normal (,...,, cos(α), sin(α)) that divides Ω into two convex sets Ω α and Ω α of equal volume. Since I(α) = I(α + π), where I(α) = u(x)dx, Ω α by continuity there is α such that I(α ) =. Applying this procedure recursively to each of the parts Ω α and Ω α, we are able to subdivide Ω into convex sets Ω i such that each of the sets is contained between two parallel hyperplanes with normal of the form (,...,, cos(β), sin(β)) at distance at most δ, and the average of u vanishes on each of them. 3

6 Consider one of these sets. By rotating the coordinate system we can assume that the normal of the enclosing hyperplanes is (,...,, 1). In these coordinates we apply the above arguments using hyperplanes with normals of the form (,...,, cos(α), sin(α), ). Continuing this procedure we end up with the desired decomposition of Ω. Theorem 3.2 Let Ω R n be a convex domain with diameter d. Then u L 2 (Ω) d π u L 2 (Ω) for all u H 1 (Ω) satisfying u(x)dx =. Ω Proof. Let us first assume that u is twice continuously differentiable. According to the previous Lemma 3.1 we are able to decompose Ω into convex subsets Ω i such that for each Ω i there is a rectangular coordinate system in which Ω i is contained in {(x, y) R n : x d i, y j δ for j =1,...,n 1}. We may assume that the interval [,d i ]onthex-axisiscontainedinω i.let R(t) bethe(n 1)-volume of the intersection of Ω i with the hyperplane x = t. In polar coordinates R(t) can be written in the form ρ(t,ω) R(t) = r n 2 dr dω = 1 ρ n 1 (t, ω)dω, S n 2 n 1 S n 2 where ρ(t, ω) is the distance of the boundary point of Ω i at (t, ω) tothe x-axis. Since Ω i is convex, ρ is a concave function with respect to t. 1 From the smoothness of u it can be seen that there are constants c 1, c 2 and c 3 such that u(x, y)dxdy u(x, )R(x)dx c 1 Ω i δ (3.1) Ω i u Ω i x (x, y) 2 dx dy u x (x, ) 2 R(x)dx c 2 Ω i δ (3.2) u(x, y) 2 dx dy u(x, ) 2 R(x)dx c 3 Ω i δ (3.3) Ω i Let ω S n 2. Since u(, ) H 1 (,d i ), we can apply Lemma 2.1 to û ω (x) :=u(x, ) u ω,where u ω := u(x, )ρn 1 (x, ω)dx. ρn 1 (x, ω)dx 1 In [5] it is claimed that R(t) is a concave function, which is not true for n 3. 4

7 Hence, û ω (x) 2 ρ n 1 (x, ω)dx d2 di i u π 2 x (x, ) 2 ρ n 1 (x, ω)dx. Applying Fubini s theorem we obtain d 2 i u π 2 x (x, ) 2 R(x)dx = 1 n 1 1 n 1 S n 2 = 1 n 1 S n 2 di d 2 i u π 2 x (x, ) 2 ρ n 1 (x, ω)dxdω û ω (x) 2 ρ n 1 (x, ω)dx dω û ω (x)u(x, )ρ n 1 (x, ω)dxdω S n 2 u(x, ) 2 R(x)dx M u(x, )R(x)dx, where M := max ω S n 2 u ω.by Ω i u(x, y)dxdy =, (3.1) and (3.2) we are lead to u(x, ) 2 R(x)dx d2 i π 2 d2 i π 2 u x (x, ) 2 R(x)dx + c 1 Ω i Mδ ( ) u(x, y) 2 d 2 i dx dy + c 1 M + c 2 Ω Ω i π 2 i δ. From (3.3) and the summation over i we obtain u(x, y) 2 dx dy d2 u(x, y) 2 d 2 dx dy +(c Ω π 2 1 M + c 2 Ω π + c 3) Ω δ 2 and, since δ>isarbitrary, the desired estimate is proven. The assertion follows from the density of C (Ω) in H 1 (Ω). Acknowledgement Thanks go to C. Carstensen for pointing out the mistake in [5]. References [1] G. Acosta, and R. G. Durán: An optimal Poincaré inequalityinl 1 for convex domains. Preprint, University of Buenos Aires, 23. [2] M. Bebendorf, and W. Hackbusch: Existence of H-Matrix Approximants to the Inverse FE-Matrix of Elliptic Operators with L -Coefficients. to appear in Numer. Math., 23. 5

8 [3] R. Courant, and D. Hilbert: Methoden der mathematischen Physik. I, Dritte Auflage, Springer, Berlin, [4] H. G. Eggleston: Convexity. Cambridge Univ. Press, New York, [5] L. E. Payne, and H. F. Weinberger: An optimal Poincaré inequality for convex domains. Arch. Rational Mech. Anal. 5, , 196. [6] J. Wloka: Partial differential equations, Translated from the German by C. B. Thomas and M. J. Thomas, Cambridge Univ. Press, Cambridge, [7] R. Verfürth: A note on polynomial approximation in Sobolev spaces. M2ANMath.Model.Numer.Anal.33, no. 4, ,