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

Kyeong-Hun Kim ()
Additional contact information
Kyeong-Hun Kim: Korea University

Journal of Theoretical Probability, 2014, vol. 27, issue 1, 107-136

Abstract: Abstract In this article, we study parabolic stochastic partial differential equations (see Eq. (1.1)) defined on arbitrary bounded domain $$\mathcal{O }\subset \mathbb{R }^d$$ admitting the Hardy inequality 0.1 $$\begin{aligned} \int _{\mathcal{O }}|\rho ^{-1}g|^2\,\text{ d}x\le C\int _{\mathcal{O }}|g_x|^2 \text{ d}x, \quad \forall g\in C^{\infty }_0(\mathcal{O }), \end{aligned}$$ where $$\rho (x)=\text{ dist}(x,\partial \mathcal{O }).$$ Existence and uniqueness results are given in weighted Sobolev spaces $$\mathfrak{H }_{p,\theta }^{\gamma }(\mathcal{O },T),$$ where $$p\in [2,\infty ), \gamma \in \mathbb{R }$$ is the number of derivatives of solutions and $$\theta $$ 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.

Keywords: Hardy inequality; Stochastic partial differential equation; Non-smooth domain; $$L_p$$ -theory; Weighted Sobolev space; 60H15; 35R60 (search for similar items in EconPapers)
Date: 2014
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10959-012-0459-7 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:27:y:2014:i:1:d:10.1007_s10959-012-0459-7

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1007/s10959-012-0459-7

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().