Regularity estimates do not create boundary compatibility

Under the choice assumptions of the cited Sobolev trace interfaces (the Axiom of Choice), the global $H^2$ and higher-order boundary theorems of this page (Global $H^2$ Dirichlet regularity, Higher-order boundary regularity for Dirichlet problems) take the solution in $H_0^1(\mathrm{\Omega })$, equivalently with zero trace, or apply after subtracting a lifting of the same Sobolev order as the regularity sought, with the resulting forcing in the required data space. An $H^1$ lifting alone does not supply an $H^2$ or higher-order estimate. They are a priori estimates, not existence or compatibility statements: they cannot manufacture boundary regularity for a datum that is not the trace of an $H^s$ function. On a nonsmooth domain with a corner, smooth coefficients and boundary data on each open boundary piece do not remove the corner obstruction. In the inhomogeneous weak Dirichlet problem the datum must lie in the trace range and be lifted before the estimates apply (Weak Dirichlet solutions for a divergence-form operator, The inhomogeneous weak Dirichlet problem by a trace lifting); the companion examples of this pair's examples page show two smooth boundary pieces with no solution continuous on the closure, and show that the $C^2$ boundary hypothesis itself cannot be dropped. No proof is supplied here; the remark records the scope boundary of the estimates.

The hypotheses of Hunter's Theorems 4.30-4.31 (printed pp. 114-116) include zero Dirichlet data, or data handled by a lifting; Teschl's Example 10.1 (printed p. 242) exhibits the reentrant-corner obstruction to the $C^2$ boundary hypothesis. The two companion examples are referenced here in prose rather than by dependency, because the estimates are the A-page content and the examples record failure modes only.