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Inventiones mathematicae

, Volume 74, Issue 3, pp 441–448 | Cite as

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

  • Elliott H. Lieb
Article

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,Bx, ofB such that λ(ABx)<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 (ABx)} in terms of λ(A), whenB is a ball; (ii) A compactness lemma for certain sequences inW1,p(ℝ n ).

Keywords

Boundary Condition Bounded Domain Dirichlet Boundary Dirichlet Boundary Condition Lower Eigenvalue 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Elliott H. Lieb
    • 1
  1. 1.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA

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