Complete Continuity of Eigen-Pairs of Weighted Dirichlet Eigenvalue Problem

Abstract

In this paper, we will study the dependence of eigen-pairs $$(\lambda _k(\rho ), \varphi _k(x,\rho ))$$ of weighted Dirichlet eigenvalue problem on weights $$\rho $$ . It will be shown that $$\lambda _k(\rho )$$ and $$\varphi _k(x,\rho )$$ are completely continuous (CC) in $$\rho $$ . Precisely, when $$\rho _n$$ is weakly convergent to $$\rho $$ in some Lebesgue space, $$\lambda _k(\rho _n)$$ is convergent to $$\lambda _k(\rho )$$ . As for the convergence of eigenfunctions, since eigenvalues may have multiple eigenfunctions, it will be shown that the distance from $$\varphi _k(x,\rho _n)$$ to the eigen space $$V_k(\rho )$$ of $$\lambda _k(\rho )$$ is tending to zero. As applications, the CC dependence of solutions of linear inhomogeneous equations and the CC dependence of the heat kernels on coefficients will be given.