Continuity estimates for n-harmonic equations

We investigate the nonhomogeneous n-harmonic equation div|∇u| ^n-2∇u = f for u in the Sobolev space ^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 ℋ¹ (Ω), so in particular, if f ∈ 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 log^α ℒ(Ω) into ℒⁿ 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. Indiana University Mathematics Journal