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On the lowest eigenvalue of the Laplacian for the intersection of two domains

Authors:	Elliott H Lieb

Institution:	(1) Departments of Mathematics and Physics, Princeton University, P.O.B. 708, 08544 Princeton, NJ, USA

Abstract:	IfA andB are two bounded domains in _n and (A), (B) are the lowest eigenvalues of – with Dirichlet boundary conditions then there is some translate,B _x, ofB such that (AB _x)<(A)+(B). A similar inequality holds for $\lambda _p (A) = \inf \{ \parallel \nabla f\parallel _p^p /\parallel f\parallel _p^p \|f \in W_0^{1,p} (A)\}$ .There are two corollaries of this theorem: (i) A lower bound for sup_x {volume (AB _x)} in terms of (A), whenB is a ball; (ii) A compactness lemma for certain sequences inW ^1,p(_n).Work partially supported by U.S. National Science Foundation grant PHY-8116101 A01. AMS(MOS) Classification: 35P15

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