On \(L_p\)-theory of stochastic partial differential equations of divergence form in \(C^1\) domains
type: Item
instance of: scholarly article
title: On \(L_p\)-theory of stochastic partial differential equations of divergence form in \(C^1\) domains
zbMATH Open document ID: 1066.60063
author: Kyeong-Hun Kim
publication date: 2005-02-22T00:00:00Z
review text: The author deals with an \(L^p\)-theory of stochastic partial differential equations of the form \[ du=(D_i(a^{ij}u_{x^j}+b^iu+f^i)+\bar{b}^i u_{x^i}+c u+ \bar{f})dt+(\sigma^{ik}u_{x^i} + \nu^k u+g^k)dw^k_t, \] where \(x\) belongs to some open set in \(\mathbb{R}^d\), \(t\geq 0\), and \((w^k_t)_{k\geq 1}\) are independent one-dimensional Wiener processes. The coefficients of the above equation are random functions depending both on the time and space variable. Existence and uniqueness of solution in a certain stochastic Banach space is studied. The techniques are based on Sobolev spaces with weights allowing the derivatives of the solutions to blow up near the boundary. Hölder continuity of the solution in \(t\) and \(x\) is also studied. The main motivation of this setting comes from \textit{N. V. Krylov} and \textit{S. V. Lototsky} [SIAM J. Math. Anal. 31, No. 1, 19--33 (1999; Zbl 0943.60047)].
reviewed by: Lluís Quer-Sardanyons
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35R60
zbMATH DE Number: 2137176
zbMATH Keywords: Sobolev spaces with weights
MaRDI profile type: Publication
full work available at URL: s00440-004-0368-5
OpenAlex ID: W1997384891
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DOI: 10.1007/S00440-004-0368-5
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
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description: scientific article; zbMATH DE number 2137176