Continuity estimates for $n$-harmonic equations

Tadeusz Iwaniec, Jani Onninen
2007 Indiana University Mathematics Journal  
We investigate the nonhomogeneous n-harmonic equation div 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 H 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
more » ... hese 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.
doi:10.1512/iumj.2007.56.2987 fatcat:omv7is4jdzdmfftdefun4b4oum