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

The functionalis bounded on $H_0^1(\Omega )$ by the Cauchy-Schwarz inequality and the Poincare inequality:The Riesz representation theorem therefore supplies a unique $u\in H_0^1(\Omega )$ satisfyingfor every test function. This is exactly the weak identity from part (a). Uniqueness also follows by testing the homogeneous difference with itself.