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

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- Solution a

Solution

The zero-boundary Sobolev space is

H_{0}^{1} (Ω) = \overline{C_{c}^{\infty} (Ω)}^{H^{1}} .

On it define

(u, v)_{H_{0}^{1}} = \int_{Ω} \nabla u \cdot \nabla v .

If this quadratic form vanishes, then

\nabla u = 0

. The Poincare inequality gives

∥ u ∥_{L^{2}} \leq C ∥\nabla u ∥_{L^{2}} = 0,

so

u = 0

. It is therefore an inner product, and its norm is equivalent to the usual

H^{1}

u \in H_{0}^{1} (Ω)

is a weak solution when

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

This follows from integration by parts and incorporates the homogeneous Dirichlet boundary condition through membership in

H_{0}^{1}

.

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