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Continuity estimates for $n$-harmonic equations
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
doi:10.1512/iumj.2007.56.2987
fatcat:omv7is4jdzdmfftdefun4b4oum