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

Elliott H. Lieb

doi:10.1007/BF01394245

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

Elliott H. Lieb

Mathematics

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Abstract

If A and B 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, of B such that λ(A∩B_x)<λ(A)+λ(B). A similar inequality holds for {Mathematical expression}.There are two corollaries of this theorem: (i) A lower bound for sup_x {volume (A∩B_x)} in terms of λ(A), when B is a ball; (ii) A compactness lemma for certain sequences in W^{1, p}(ℝ_n).

Original language	English (US)
Pages (from-to)	441-448
Number of pages	8
Journal	Inventiones Mathematicae
Volume	74
Issue number	3
DOIs	https://doi.org/10.1007/BF01394245
State	Published - Oct 1983

All Science Journal Classification (ASJC) codes

General Mathematics

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N2 - If A and B are two bounded domains in ℝn and λ(A), λ(B) are the lowest eigenvalues of -Δ with Dirichlet boundary conditions then there is some translate, Bx, of B such that λ(A∩Bx)<λ(A)+λ(B). A similar inequality holds for {Mathematical expression}.There are two corollaries of this theorem: (i) A lower bound for supx {volume (A∩Bx)} in terms of λ(A), when B is a ball; (ii) A compactness lemma for certain sequences in W1, p(ℝn).

AB - If A and B are two bounded domains in ℝn and λ(A), λ(B) are the lowest eigenvalues of -Δ with Dirichlet boundary conditions then there is some translate, Bx, of B such that λ(A∩Bx)<λ(A)+λ(B). A similar inequality holds for {Mathematical expression}.There are two corollaries of this theorem: (i) A lower bound for supx {volume (A∩Bx)} in terms of λ(A), when B is a ball; (ii) A compactness lemma for certain sequences in W1, p(ℝn).

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

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