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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)?

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I asked this question on Mathoverflow a week ago and did not get any answers: https://mathoverflow.net/questions/126950/boundary-regularity-of-weighted-p-laplace-equation

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Tommi
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  • 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
  • @RayYang To my knowledge the solutions for the unweighted p-Laplace are smooth except where the first derivative vanishes. I do not know if this hold for the weighted equation (I suppose it does up to the regularity allowed by the weight) or if it holds on the boundary. I'll check the references, thanks. – Tommi Apr 18 '13 at 19:10
  • Tommi, where is this proved? Is this a result you get by simply differentiating the p-Laplacian? – Ray Yang Apr 20 '13 at 20:49
  • @RayYang I don't know where this is proved, but I don't see any obstructions to it and my advisor believes it to be true. It might be in Nonlinear potential theory by Heinonen, Kilpeläinen and Martio, but maybe not (I don't have access to the book right now). The lecture notes by P.Lindqvist give a reference for the constant weight case. – Tommi Apr 22 '13 at 09:22
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    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
  • Hi Tommi, did you find what you needed? I read through some of Ladyzhenskaya and Uraltseva more carefully for another question on Math.SE, and I think I can provide a more thorough answer if you need one, but I think we've hashed out all the main points. – Ray Yang May 01 '13 at 13:49
  • Hello @RayYang,I returned home recently to discover that the book of Ladyzhenskaja and Uraltseva is currently loaned from the library. I will take a look once I can loan it; the need is not urgent, at least now. Thank you for your help. – Tommi May 06 '13 at 05:38

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