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 can successfully obtain several isoperimetric inequalities for the first and the second Dirichlet eigenvalues, the first nonzero Neumann eigenvalue of the Witten-Laplacian on bounded domains in space forms. These spectral isoperimetric inequalities extend those classical ones (i.e. the Faber-Krahn inequality, the Hong-Krahn-Szegő inequality and the Szegő-Weinberger inequality) of the Laplacian.

33 pages. Some revisions to v2. By the way, a slightly different version will appear in Journal of Spectral Theory. arXiv admin note: text overlap with arXiv:2403.08070