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


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 ).


Boundary Condition Bounded Domain Dirichlet Boundary Dirichlet Boundary Condition Lower Eigenvalue 
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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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