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A weak solution is a function $u\in H_0^1(U)$ satisfyingThe zero boundary condition is encoded by membership in $H_0^1(U)$.

On the Hilbert space $H_0^1(U)$ defineThe form is bounded and coercive:The functional ${\ell }_f$ is bounded by the Cauchy-Schwarz inequality. The Lax-Milgram theorem therefore gives a unique weak solution and a boundThus $T:L^2(U)\to H_0^1(U)$ is bounded. Composing it with the compact Rellich-Kondrashov compactness theorem for H01 embeddingshows that $T:L^2(U)\to L^2(U)$ is the compact massive-Laplacian resolvent.