{
  "id": "1508.03402",
  "version": "v1",
  "published": "2015-08-14T02:04:24.000Z",
  "updated": "2015-08-14T02:04:24.000Z",
  "title": "A counterexample to maximal $L_p$-regularity of the stochastic heat equation in polygons: the case $p>4$",
  "authors": [
    "Kyeong-Hun Kim"
  ],
  "categories": [
    "math.PR"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-14T02:04:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic heat equation",
      "counterexample",
      "regularity",
      "one-dimensional wiener process",
      "similar statement holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150803402K"
    }
  }
}