Weak Dirichlet problem for the massive Laplacian

For a bounded open $U\subset {\mathbb{R}}^n$, $m^2>0$, and $f\in L^2(U)$, there is a unique $u\in H_0^1(U)$ such thatfor every $v\in H_0^1(U)$.

The solution operator $(-\Delta +m^2{)}^{-1}:L^2(U)\to L^2(U)$ is compact on every bounded open set because it maps boundedly into $H_0^1(U)$ and the Rellich-Kondrashov compactness theorem for H01 embeds that space compactly into $L^2(U)$.