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

Elliott Lieb

Research output: Contribution to journalArticle

128 Scopus citations

Abstract

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).

Original languageEnglish (US)
Pages (from-to)441-448
Number of pages8
JournalInventiones Mathematicae
Volume74
Issue number3
DOIs
StatePublished - Oct 1 1983

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Fingerprint Dive into the research topics of 'On the lowest eigenvalue of the Laplacian for the intersection of two domains'. Together they form a unique fingerprint.

  • Cite this