Failure of a coercive Dirichlet estimate under degeneracy
= Failure of a coercive Dirichlet estimate under degeneracy
For $P=\partial_1^2+\partial_2^2$ in a three-dimensional domain, let $u_k=\phi(x_1,x_2)\chi(x_3)\sin(kx_3)$ with fixed smooth compact support. Then $\|Pu_k\|_2$ remains bounded whereas $\|u_k\|_{H^1}$ grows proportionally to $k$. Thus no uniform estimate $\|u\|_{H^1}\le C\|Pu\|_2$ holds. The missing third-direction coercivity, rather than boundary data, causes the failure.