Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 12 / 2 / i

Solution

Write

H^{1} = W^{1, 2}

and

H_{0}^{1} = W_{0}^{1, 2}

u - ψ \in H_{0}^{1} (Ω)

; this definition remains meaningful even without regularity assumptions on the boundary. For bounded full coefficients, a weak solution is a function satisfying

u \in H^{1} (Ω), u - ψ \in H_{0}^{1} (Ω), B (u, φ) = - \int_{Ω} f φ for every φ \in H_{0}^{1} (Ω),

where

B (u, φ) = \int_{Ω} a_{ij} D_{j} u D_{i} φ - \int_{Ω} q u φ .

D_{i} (a_{ij} D_{j} u) + q u = f

. No classical derivatives of the measurable coefficients are taken. Equivalently one may first test against

C_{c}^{\infty} (Ω)

and extend by density, provided this bilinear form is bounded on

H^{1} \times H_{0}^{1}

. All integrals in this definition must exist: as explained below, the printed quadratic bounds alone do not guarantee this when

A = (a_{ij})

is nonsymmetric.

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