A weighted Sobolev space theory of parabolic stochastic PDEs on non-smooth domains

2011/09/22 by Kyeong-Hun Kim, Kim, Kyeong-Hun · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Nonlinear Partial Differential Equations #Probability (math.PR) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1109.4727

openalex publication_date 2011/09/22 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

In this paper we study parabolic stochastic partial differential equations defined on arbitrary bounded domain \cO ⊂ \bR^d allowing Hardy inequality: ∫_\cO|ρ^-1g|² dx≤ C∫_\cO|g_x|² dx, ∀ g∈ C^∞₀(\cO), where ρ(x)=dist(x,∂ \cO). Existence and uniqueness results are given in weighted Sobolev spaces \frH^γ_p,θ(\cO,T), where p∈ [2,∞), γ∈ \bR is the number of derivatives of solutions and θ controls the boundary behavior of solutions. Furthermore several Hölder estimates of the solutions are also obtained. It is allowed that the coefficients of the equations blow up near the boundary.

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• Functional calculus on weighted Sobolev spaces for the Laplacian on rough domains

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