Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A counterexample to maximal $L_p$-regularity of the stochastic heat
  equation in polygons: the case $p>4$   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.

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The categories of this paper are: quant-ph
10
The categories of this paper are: q-bio
27
The categories of this paper are: comp-gas
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
19
The categories of this paper are: eess
34
The categories of this paper are: chem-ph
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The categories of this paper are: solv-int
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The categories of this paper are: q-alg
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The categories of this paper are: ao-sci
21
The categories of this paper are: adap-org
37
The categories of this paper are: supr-con
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The categories of this paper are: stat
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The categories of this paper are: astro-ph
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The categories of this paper are: cs
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The categories of this paper are: funct-an