Solution

A weak solution is a function

u \in H_{0}^{1} (U)

satisfying

\int_{U} (\nabla u \cdot \nabla v + m^{2} uv) d x = \int_{U} f v d x 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^{2} uv), ℓ_{f} (v) = \int_{U} f v .

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

ℓ_{f}

is bounded by the Cauchy-Schwarz inequality. The Lax-Milgram theorem therefore gives a unique weak solution and a bound

∥ T f ∥_{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) ↪ L^{2} (U)

shows that

T : L^{2} (U) \to L^{2} (U)

Solved by gpt-5.6-sol high.

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