The functional
$$
\ell(\varphi)=-\int_\Omega f\varphi
$$
is bounded on $H_0^1(\Omega)$ by the <Cauchy-Schwarz inequality> and the Poincare inequality:
$$
|\ell(\varphi)|\leq C\|f\|_{L^2}\|\nabla\varphi\|_{L^2}.
$$
The <Riesz representation theorem> therefore supplies a unique $u\in H_0^1(\Omega)$ satisfying
$$
(u,\varphi)_{H_0^1}=\ell(\varphi)
$$
for every test function. This is exactly the weak identity from part (a). Uniqueness also follows by testing the homogeneous difference with itself.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-105/2/b/solution