= Solution
A weak solution is a function $u\in H_0^1(U)$ satisfying
$$
\boxed{
\int_U\left(\nabla u\cdot\nabla v+m^2uv\right)\,dx
=\int_Ufv\,dx
\quad\text{for every }v\in H_0^1(U).}
$$
The zero boundary condition is encoded by membership in $H_0^1(U)$.
On the Hilbert space $H_0^1(U)$ define
$$
a(u,v)=\int_U(\nabla u\cdot\nabla v+m^2uv),
\qquad
\ell_f(v)=\int_Ufv.
$$
The form is bounded and coercive:
$$
a(u,u)=\|\nabla u\|_2^2+m^2\|u\|_2^2
\geq\min(1,m^2)\|u\|_{H^1}^2.
$$
The functional $\ell_f$ is bounded by the <Cauchy-Schwarz inequality>. The <Lax-Milgram theorem> therefore gives a unique weak solution and a bound
$$
\|Tf\|_{H_0^1}\leq C\|f\|_2.
$$
Thus $T:L^2(U)\to H_0^1(U)$ is bounded. Composing it with the compact <Rellich-Kondrashov compactness theorem for H01> embedding
$$
H_0^1(U)\hookrightarrow L^2(U)
$$
shows that $T:L^2(U)\to L^2(U)$ is the <compact massive-Laplacian resolvent>.
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