Continuity of the first eigenvalue for a family of degenerate eigenvalue problems

Abstract

For each ${\alpha\in[0,2)}$ we consider the eigenvalue problem ${-{\rm div}(|x|^\alpha \nabla u)=\lambda u}$ in a bounded domain ${\Omega\subset \mathbb{R}^N}$ (${N\geq 2}$) with smooth boundary and ${0\in \Omega}$ subject to the homogeneous Dirichlet boundary condition. Denote by ${\lambda_1(\alpha)}$ the first eigenvalue of this problem. Using ${\Gamma}$-convergence arguments we prove the continuity of the function ${\lambda_1}$ with respect to ${\alpha}$ on the interval ${[0,2)}$.