On the lowest eigenvalue of the Laplacian for the intersection of two domains (Q793226)

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 \(W^{1,p}({\mathbb{R}}^ n)\), \(1<p<\infty\).