ON THE LOWEST EIGENVALUE OF THE FRACTIONAL LAPLACIAN FOR THE INTERSECTION OF TWO DOMAINS

Abstract

AbstractWe extend a result of Lieb [‘On the lowest eigenvalue of the Laplacian for the intersection of two domains’, Invent. Math.74(3) (1983), 441–448] to the fractional setting. We prove that if A and B are two bounded domains in $\mathbb R^N$ and $\lambda _s(A)$ , $\lambda _s(B)$ are the lowest eigenvalues of $(-\Delta )^s$ , $0<s<1$ , with Dirichlet boundary conditions, there exists some translation $B_x$ of B such that $\lambda _s(A\cap B_x)< \lambda _s(A)+\lambda _s(B)$ . Moreover, without the boundedness assumption on A and B, we show that for any $\varepsilon>0$ , there exists some translation $B_x$ of B such that $\lambda _s(A\cap B_x)< \lambda _s(A)+\lambda _s(B)+\varepsilon .$