Continuity estimates for $n$-harmonic equations

Abstract

We investigate the nonhomogeneous n-harmonic equation div|∇u| n-2 ∇u = f for U in the Sobolev space W 1,n (Ω), where f is a given function in the Zygmund class L log α L(Ω). In dimension n = 2 the solutions are continuous whenever f lies in the Hardy space Η 1 (Ω), so in particular, if f ∈ L log L(Ω). We show in higher dimensions that within the Zygmund classes the condition α > n - 1 is both necessary and sufficient for the solutions to be continuous. We also investigate continuity of the map f → ∇u, from L logα L(Ω) into L n log β L(Ω), for -1 < β < nα n-1 -1. These and other results of the present paper, though anticipated by simple examples, are in fact far from routine. Certainly, they are central in the p-harmonic theory.