A minimax formula for the principal eigenvalues of Dirichlet problems and its applications

Abstract

A minimax formula for the principal eigenvalue of a nonselfadjoint Dirichlet problem was established in [8,18]. In this paper we generalize this formula to the case where an indefinite weight is present. Our proof requires less regularity and, unlike that in [8,18], does not rely on semigroups theory nor on stochastic differential equations. It makes use of weighted Sobolev spaces. An application is given to the study of the uniformity of the antimaximum principle.

Keywords

• Nonselfadjoint elliptic problem
• principal eigenvalue
• indefinite weight
• minimax formula
• weighted Sobolev spaces
• degenerate elliptic equations
• antimaximum principle.