L^p regularity of least gradient functions

2019/04/24 by Wojciech Górny, Górny, Wojciech
Mathematics · #35J20 #35J25 #35J75 #35J92 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP #msc:35J20 #msc:35J25 #msc:35J75 #msc:35J92

paper · pdf · doi:10.48550/arxiv.1904.11348

arXiv admin note: text overlap with arXiv:1811.11138

arxiv created 2019/04/24 · openalex publication_date 2019/04/24 · arxiv updated 2019/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is shown that solutions to the anisotropic least gradient problem for boundary data f ∈ L^p(∂Ω) lie in L^(Np)/(N-1)(Ω); the exponent is shown to be optimal. Moreover, the solutions are shown to be locally bounded with explicit bounds on the rate of blow-up of the solution near the boundary in two settings: in the anisotropic case on the plane and in the isotropic case in any dimension.

Citations

• Existence of minimisers in the least gradient problem for general boundary data

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