# Boundary regularity of weighted p-Laplace equation

It seems that I need to find some regularity results for the weighted p-Laplace equation, namely $\nabla \cdot (\gamma |\nabla u|^{p-2} \nabla u) = 0$ with Dirichlet boundary values.

Suppose that we have a smooth bounded domain $\Omega$ and smooth Dirichlet boundary values. Take the weight $\gamma$ to be smooth and positive in the closure of $\Omega$. Take p in the interval $]1,\infty[.$

What can we say about the regularity of the solution u on the boundary $\partial \Omega$?

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Some references I have found:

An article by Xiangling Fan titled "Global $C^{1,\alpha}$ regularity for variable exponent elliptic equations in divergence form" states that with certain assumptions on the domain, weight and boundary conditions we have $u \in C^{1,\alpha}(\overline{\Omega})$. The article focuses on the variable exponent case, though, so better results might exist for static $p$.

The work of Kaj Nyström et al concerns the boundary behaviour of the p-laplace equation, but they assume the solution is everywhere positive and zero near the part of boundary they are investigating. These assumptions are too strict for my purposes.

The lecture notes of Peter Lindqvist seem to focus more on interior regularity and non-weighted case.

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I also asked this on math.SE: http://math.stackexchange.com/questions/364404/regularity-of-weighted-p-laplace-equation-up-to-boundary

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