Failure of zero extension in parabolic Sobolev spaces

Abstract

We show that spatial zero extension across the boundary may fail in parabolic Sobolev spaces ${\mathring{\mathcal{H}}}_p^1((0,T)\times \Omega )$, which can also be characterized as $$L_p(0,T;{\mathring{W}}_p^1(\Omega ))\cap W_p^1(0,T;W_p^{-1}(\Omega )).$$ More precisely, for any $p\in [1,\infty )$, we construct a function $u\in {\mathring{\mathcal{H}}}_p^1((0,T)\times {\mathbb{R}}_{+}^d)$ whose zero extension does not belong to ${\mathcal{H}}_p^1((0,T)\times {\mathbb{R}}^d)$. The obstruction occurs even for a flat boundary and is caused by a self-similar boundary layer concentrated at the initial-boundary corner, which produces a boundary supported normal flux defect after zero extension. We also discuss the suitability of various Sobolev-type spaces as solution spaces for parabolic equations in divergence form.

arXiv comment18 pages