Formula omitted] bounds of eigenfunctions with an indefinite weight function

Abstract

This paper is concerned with the L∞ bounds of eigenfunctions to the eigenvalue problem Δu=−λm(x)uin Ω,u=0on ∂Ω,where Ω is a bounded domain and m(x) is an indefinite weight function. For the Dirichlet eigenvalue problem when m(x)≡1, the L∞ bound of eigenfunction is obtained by classical heat kernel estimate. However, the case is quite different for the weighted problem, additional difficulty appears if m(x) is indefinite. In this paper, we investigate the L∞ bounds of eigenfunctions by the means of comparison argument and fundamental solution of heat equation.