Global BMO-Sobolev Estimates for Second-Order Linear Elliptic Equations on Lipschitz Domains

2024/09/29 by Hongjie Dong, Dachun Yang, Dong, Hongjie +3
Mathematics · #Advanced Mathematical Physics Problems #Nonlinear Partial Differential Equations #Differential Equations and Boundary Problems

paper · pdf · doi:10.48550/arxiv.2409.19498

Abstract

Let n ≥ 2 and Ω⊂ ℝⁿ be a bounded Lipschitz domain. In this article, we establish first-order global regularity estimates in the scale of BMO spaces on Ω for weak solutions to the second-order elliptic equation div(A ∇ u) = div , \boldsymbolf in Ω. This is achieved under minimal regularity assumptions on Ω and the coefficient matrix A, utilizing the pointwise multiplier characterization of the BMO space on Ω. As an application, we also obtain global estimates of ∇ u in the Lebesgue space L¹(Ω) when \boldsymbolf belongs to the Hardy space on Ω.

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