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Let $\Omega\subset\mathbb R^n$, $u\in W^{1,p}(\Omega)$, $p>1$ satisfies $$\int_{\Omega}|\triangledown u|^{p-2}\triangledown u\triangledown\phi=0~~~,\forall\phi\in C_0^{\infty}(\Omega)$$ Then we call $u$ harmonic function.

  • When $p=n$, prove $u\in C^\alpha(\Omega)$

  • When $p<n$, prove $u\in C^{1,\alpha}(\Omega)$

I wonder how to prove two questions. The hints say that the former is proved by Caccioppoli inequality; the latter by Moser iteration.

Any advice is helpful. Thank you.

gaoxinge
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  • Crossposted to MO: http://mathoverflow.net/questions/209744/a-problem-about-a-harmonic-function – user43208 Jun 20 '15 at 10:14
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    The functions you are considering are usually known as p-harmonic functions and the equation is the p-Laplace equation. You should start by perusing the lecture notes of Peter Lindqvist: http://www.math.ntnu.no/~lqvist/p-laplace.pdf – Tommi Jun 20 '15 at 12:06

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