Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 105 / 2 / d / Solution

A fixed smooth choice $g\equiv 0$ already gives a counterexample. Select nonzero smooth functions $\varphi (x_1,x_2)$ and $\chi (x_3)$ with product support compactly contained in a small box inside the shell. LetThen $u_k$ has zero boundary trace and is a classical, hence weak, solution. The $L^2$ norms of $f_k$ are bounded independently of $k$. On the other hand,The squared norm of $\chi \cos (k{x}_3)$ tends to $\frac{1}{2}\Vert \chi {\Vert }_2^2$, by the oscillatory integral identity for ${\cos }^2$. The other term has bounded norm, so $\Vert u_k{\Vert }_{H^1}$ grows at least proportionally to $k$. ThereforeThis failure of a coercive Dirichlet estimate under degeneracy loses control in the third direction; it does not rely on varying $g$ with the sequence.