A Weighted Sobolev Space Theory of Parabolic Stochastic PDEs on Non-smooth Domains

Kyeong Hun Kim

doi:10.1007/s10959-012-0459-7

A Weighted Sobolev Space Theory of Parabolic Stochastic PDEs on Non-smooth Domains

Kyeong Hun Kim

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

16 Citations (Scopus)

Abstract

In this article, we study parabolic stochastic partial differential equations (see Eq. (1.1)) defined on arbitrary bounded domain O ⊂ ℝ^d admitting the Hardy inequality (Formula Presented) where ρ (x)= dist(x,∂ O). Existence and uniqueness results are given in weighted Sobolev spaces (Formula Presented) where p ∈ [2,∞), γ ∈ ℝ is the number of derivatives of solutions and θ controls the boundary behavior of solutions (see Definition 2.5). Furthermore, several Hölder estimates of the solutions are also obtained. It is allowed that the coefficients of the equations blow up near the boundary.

Original language	English
Pages (from-to)	107-136
Number of pages	30
Journal	Journal of Theoretical Probability
Volume	27
Issue number	1
DOIs	https://doi.org/10.1007/s10959-012-0459-7
Publication status	Published - 2014 Mar

Keywords

Hardy inequality
L-theory
Non-smooth domain
Stochastic partial differential equation
Weighted Sobolev space

ASJC Scopus subject areas

Statistics and Probability
Mathematics(all)
Statistics, Probability and Uncertainty

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10.1007/s10959-012-0459-7

Cite this

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title = "A Weighted Sobolev Space Theory of Parabolic Stochastic PDEs on Non-smooth Domains",

abstract = "In this article, we study parabolic stochastic partial differential equations (see Eq. (1.1)) defined on arbitrary bounded domain O ⊂ ℝd admitting the Hardy inequality (Formula Presented) where ρ (x)= dist(x,∂ O). Existence and uniqueness results are given in weighted Sobolev spaces (Formula Presented) where p ∈ [2,∞), γ ∈ ℝ is the number of derivatives of solutions and θ controls the boundary behavior of solutions (see Definition 2.5). Furthermore, several H{\"o}lder estimates of the solutions are also obtained. It is allowed that the coefficients of the equations blow up near the boundary.",

keywords = "Hardy inequality, L-theory, Non-smooth domain, Stochastic partial differential equation, Weighted Sobolev space",

author = "Kim, {Kyeong Hun}",

note = "Funding Information: Acknowledgments The author is grateful to Ildoo Kim for finding several typos in the earlier version of the article, to N.V. Krylov for providing the author an example and to P.A. Cioica and F. Lindner for useful discussions regarding the numerical approximations of SPDEs on non-smooth domains. The author is also thankful to the referee for much useful advice. The research of this author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (20110015961).",

year = "2014",

month = mar,

doi = "10.1007/s10959-012-0459-7",

language = "English",

volume = "27",

pages = "107--136",

journal = "Journal of Theoretical Probability",

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AU - Kim, Kyeong Hun

N1 - Funding Information: Acknowledgments The author is grateful to Ildoo Kim for finding several typos in the earlier version of the article, to N.V. Krylov for providing the author an example and to P.A. Cioica and F. Lindner for useful discussions regarding the numerical approximations of SPDEs on non-smooth domains. The author is also thankful to the referee for much useful advice. The research of this author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (20110015961).

PY - 2014/3

Y1 - 2014/3

N2 - In this article, we study parabolic stochastic partial differential equations (see Eq. (1.1)) defined on arbitrary bounded domain O ⊂ ℝd admitting the Hardy inequality (Formula Presented) where ρ (x)= dist(x,∂ O). Existence and uniqueness results are given in weighted Sobolev spaces (Formula Presented) where p ∈ [2,∞), γ ∈ ℝ is the number of derivatives of solutions and θ controls the boundary behavior of solutions (see Definition 2.5). Furthermore, several Hölder estimates of the solutions are also obtained. It is allowed that the coefficients of the equations blow up near the boundary.

AB - In this article, we study parabolic stochastic partial differential equations (see Eq. (1.1)) defined on arbitrary bounded domain O ⊂ ℝd admitting the Hardy inequality (Formula Presented) where ρ (x)= dist(x,∂ O). Existence and uniqueness results are given in weighted Sobolev spaces (Formula Presented) where p ∈ [2,∞), γ ∈ ℝ is the number of derivatives of solutions and θ controls the boundary behavior of solutions (see Definition 2.5). Furthermore, several Hölder estimates of the solutions are also obtained. It is allowed that the coefficients of the equations blow up near the boundary.

KW - Hardy inequality

KW - L-theory

KW - Non-smooth domain

KW - Stochastic partial differential equation

KW - Weighted Sobolev space

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U2 - 10.1007/s10959-012-0459-7

DO - 10.1007/s10959-012-0459-7

M3 - Article

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SN - 0894-9840

VL - 27

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EP - 136

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

IS - 1

ER -

A Weighted Sobolev Space Theory of Parabolic Stochastic PDEs on Non-smooth Domains

Abstract

Keywords

ASJC Scopus subject areas

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