Regularity of solutions of quasi-linear elliptic equations with L\log^m\n L coefficients

2019/11/19 by Julian Edward, Edward, Julian, Steve Hudson +3
Computer Science · Mathematics · #35J62 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1911.08622

openalex publication_date 2019/11/19 · openalex created_date 2022/07/19 · openalex updated_date 2026/07/28

Abstract

Let D be an bounded region in bf Rⁿ. The regularity of solutions of a\nfamily of quasilinear elliptic partial differential equations is studied, one\nexample being \Δ_nu=Vu^n-1. The coefficients are assumed to be in the\nspace L\log^mL(D) for m>n-1. Using a Moser iteration argument coupled\nwith the Moser-Trudinger inequality, a local L^\∞ bound on the solution\nu is proven. A Harnack-type inequality is then proven. These results are\nshown to be sharp with respect to m. Then essential continuity of u is\nproven, and away from the boundary a bound on the modulus of continuity.\n

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