Sobolev space theory for Poisson's and the heat equations in non-smooth domains via superharmonic functions and Hardy's inequality

2023/04/20 by Jinsol Seo, Seo, Jinsol
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Advanced Harmonic Analysis Research

paper · pdf · doi:10.48550/arxiv.2304.10451

Abstract

We prove the unique solvability for the Poisson and heat equations in non-smooth domains Ω⊂ ℝ^d in weighted Sobolev spaces. The zero Dirichlet boundary condition is considered, and domains are merely assumed to admit the Hardy inequality: ∫_Ω|(f(x))/(d(x,∂Ω))|² d x≤ N∫_Ω|∇ f|² d x , ∀ f∈ C_c^∞(Ω) . To describe the boundary behavior of solutions, we introduce a weight system that consists of superharmonic functions and the distance function to the boundary. The results provide separate applications for the following domains: convex domains, domains with exterior cone condition, totally vanishing exterior Reifenberg domains, conic domains, and domains Ω⊂ℝ^d which the Aikawa dimension of Ω^c is less than d-2.

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