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Take the weighted p-Laplace equation, $\nabla \cdot (\gamma |\nabla u|^{p-2}\nabla u) = 0$, with smooth Dirichlet boundary values on some smooth and bounded domain. Suppose the weight $\gamma$ is also smooth in the closure of the domain, and positive. (We then have $\gamma \geq \gamma_0 > 0$ for some constant $\gamma_0$.)

In general the derivative of the solution u is Hölder-continuous in the domain and the solution u is smooth whenever its derivative does not vanish. These results hold inside the domain.

An article by Xiangling Fan titled "Global $C^{1,α}$ regularity for variable exponent elliptic equations in divergence form" states that the derivative of the solution $u$ is Hölder-continuous in the closure of the domain.

Is it known if the solution $u$ is smooth in the closure of the domain, aside from the points where its derivative vanishes? Can the derivative of the solution vanish on the boundary (assuming non-zero boundary values)?


I asked this question on Mathoverflow a week ago and did not get any answers:

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We can take $\gamma$ to be bounded below by some positive constant. I tried to imply this in the question, but clarified it now. Thank you for the comment. – Tommi Brander Apr 17 '13 at 19:38
Given that there are counterexamples for the p-Laplacian which show that for the proper values of $p$ and the spatial dimension, you cannot do better than $C^{1,\alpha}$ for interior regularity, what are you hoping to show? I think global $C^{1,\alpha}$ is the best you can do? – Ray Yang Apr 18 '13 at 17:58
Oops. I should say that the counterexamples are for $C^{1,1}$. I am thinking of the ones in J. Lewis, Smoothness of certain degenerate elliptic equations, Proc. Amer. Math. Soc. 80 (1980), 259-265. – Ray Yang Apr 18 '13 at 18:05
You might want to look at Ladyzhenskaya and U'raltseva's book on Linear and quasilinear elliptic equations, specifically the chapter on quasilinear equations with divergence structure. – Ray Yang Apr 18 '13 at 18:56
Hi Tommi, I took a look. Lindqvist's reference is to Lewis' paper "Capacitary Functions in Convex Rings," where he refers to Ladyzhenskaya and Uraltseva's book, where they prove the required theorem (Thm 6.3 in Chapter 4) by iterative application of the Schauder estimates to derivatives of $u$ in a region where $|\nabla u|$ is strictly bounded away from 0. If $\gamma$ is strictly bounded away from 0 and smooth, this procedure should work in your case also. – Ray Yang Apr 22 '13 at 17:48

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