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 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 | |
| Publication status | Published - 2014 Mar |
Bibliographical 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).
Keywords
- Hardy inequality
- L-theory
- Non-smooth domain
- Stochastic partial differential equation
- Weighted Sobolev space
ASJC Scopus subject areas
- Statistics and Probability
- General Mathematics
- Statistics, Probability and Uncertainty