Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 105 / 2 / c / Solution

After integration by parts, the weak formulation is

\int_{Ω} (\nabla u \cdot \nabla φ + u φ) = - \int_{Ω} f φ (φ \in H_{0}^{1} (Ω)) .

The left side is the inner product

(u, φ)_{*} = \int_{Ω} (\nabla u \cdot \nabla φ + u φ),

which is positive definite and induces the usual

H^{1}

norm. The right side is bounded in this norm. The Riesz representation theorem, equivalently the Lax-Milgram theorem, gives a unique weak solution. This is the weak Dirichlet problem for the massive Laplacian with mass one and the signs multiplied by

- 1

.

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