A counterexample to maximal $L_p$-regularity of the stochastic heat equation in polygons: the case $p>4$

Abstract

Let $D$ be a domain in $R^d$ and $u$ be the solution to the stochastic heat equation $$ du=\Delta u dt+ g\,dW_t, \quad t>0, x\in D, $$ with zero initial and boundary data. Here $W_t$ is a one-dimensional Wiener process on a probability space $\Omega$. It has been proved (see below for references) that for any $p\geq 2$ the inequality $$ \|\nabla u\|_{L_p(\Omega\times [0,T]\times D)} \leq c \|g\|_{L_p(\Omega\times [0,T]\times D)} $$ holds if $\partial D\in C^1$. In this note we prove that if $p>4$ then this inequality fails in any polygon in $R^2$ having an angle greater than or equal to $\frac{p\pi}{2(p-2)}$. We also show that a similar statement holds in higher dimensional polygons. The counterexample introduced here is based on personal communication with N.V. Krylov.