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On the lowest eigenvalue of the Laplacian for the intersection of two domains. (English) Zbl 0538.35058
Main result: Let $$A,B\subset {\mathbb{R}}^ n$$ be open and non-empty, and let $$\lambda(A)$$, $$\lambda(B)$$ be the infimum of the $$L^ 2$$-spectrum of $$- \Delta$$ with Dirichlet boundary conditions. Let $$B_ x$$ denote the translate of B by $$x\in {\mathbb{R}}^ n$$. Then $$\inf_{x}\lambda(A\cap B_ x)\leq \lambda(A)+\lambda(B)$$, with strict inequality if A and B are bounded. A similar inequality is proved for $$\lambda_ p(A):= \inf \{\| f\|^ p_ p/\| f\|^ p_ p;$$ $$f\in C_ 0^{\infty}(A),$$ $$f\neq 0\},$$ $$1\leq p<\infty$$. A consequence of this result is a lower bound for $$\sup_{x}vol(A\cap B_ x)$$ in terms of $$\lambda(A)$$, when B is a ball. This result was motivated by the isoperimetric inequality $$\lambda(A)\geq \beta_ n$$ $$vol(A)^{-2/n}$$ where $$\beta_ n$$ is the lowest eigenvalue for a ball of unit volume. A second consequence is a compactness lemma for certain sequences in $$W^{1,p}({\mathbb{R}}^ n)$$, $$1<p<\infty$$.
Reviewer: J.Voigt

##### MSC:
 35P15 Estimates of eigenvalues in context of PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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##### References:
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