A Local Lewy Theorem for p-Harmonic Function with Non-zero Gradient in R3
Jiahuan Li, Yilu Liu, Xi-Nan Ma
Abstract
We establish a local Lewy-type theorem for \(p\)-harmonic function with non-zero gradient in dimension three space R3. Let \(1<p<∞\), and let \(u∈ W1,ploc(Ω)\) be a weak \(p\)-harmonic function in a domain \(Ω⊂ R3\), assume it satisfies \(|Du|>0\), we prove that a locally homeomorphic gradient map \(Du\) must have non-vanishing Hessian determinant. Hence \(Du\) is a local diffeomorphism.
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