Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 105 / 3 / i / Solution

In three dimensions the Sobolev embedding theorem givesConsequently, for $w\in H^2(U)\cap H_0^1(U)$,Thus $f+|Dw{|}^{2}w\in L^2(U)$. The Dirichlet Poisson regularity theorem on a bounded $C^2$ domain says thathas a unique $v\in H^2(U)\cap H_0^1(U)$ andHence $\Phi (w)=v$ is well defined.