Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 105 / 3 / b / i

Table of contents
- Solution i

Solution

H = H_{0}^{1} (U)^{2}

. A weak solution is a vector

ϕ \in H

such that

\int_{U} (D ϕ : D v + (A ϕ) \cdot v) = ⟨ F, v ⟩ for every v \in H,

where

D ϕ : D v = \sum_{j = 1}^{2} D ϕ_{j} \cdot D v_{j}

and the natural data space is

F \in H^{- 1} (U)^{2} = (H_{0}^{1} (U)^{2})^{*}

.

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