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

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Abstract

IfA andB are two bounded domains in ℝ n and λ(A), λ(B) are the lowest eigenvalues of −Δ with Dirichlet boundary conditions then there is some translate,B x, ofB such that λ(A∩B x)<λ(A)+λ(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 (A∩B x)} in terms of λ(A), whenB is a ball; (ii) A compactness lemma for certain sequences inW 1,p(ℝ n ).

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

    Elliott H. Lieb

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  1. Elliott H. Lieb
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Work partially supported by U.S. National Science Foundation grant PHY-8116101 A01. AMS(MOS) Classification: 35P15

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Lieb, E.H. On the lowest eigenvalue of the Laplacian for the intersection of two domains. Invent Math 74, 441–448 (1983). https://doi.org/10.1007/BF01394245

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