Several isoperimetric inequalities of Dirichlet and Neumann eigenvalues of the Witten Laplacian

Abstract

In this paper, by mainly using the rearrangement technique and suitably constructing trial functions, under the constraint of fixed weighted volume, we successfully obtain several isoperimetric inequalities for the first and the second Dirichlet eigenvalues, the first non-zero Neumann eigenvalue of the Witten Laplacian on bounded domains in space forms. These spectral isoperimetric inequalities extend the classical ones (i.e., the Faber–Krahn inequality, the Hong–Krahn–Szegő inequality, and the Szegő–Weinberger inequality) of the Laplacian.