 Failure of a coercive Dirichlet estimate under degeneracy

For $P={\partial }_1^2+{\partial }_2^2$ in a three-dimensional domain, let $u_k=\varphi (x_1,x_2)\chi (x_3)\sin (k{x}_3)$ with fixed smooth compact support. Then $\Vert P{u}_k{\Vert }_2$ remains bounded whereas $\Vert u_k{\Vert }_{H^1}$ grows proportionally to $k$. Thus no uniform estimate $\Vert u{\Vert }_{H^1}\le C\Vert Pu{\Vert }_2$ holds. The missing third-direction coercivity, rather than boundary data, causes the failure.