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We know that, in general, the 'best' regularity of p-harmonic functions is $C^{1,\alpha}$, $0<\alpha<1$.

Recently, I saw a method of regularized problems as follows: For each $\epsilon>0$, consider the problem in a domain $\Omega$ (in my question a planar domain) \begin{equation} (1)\qquad \left\lbrace \begin{array}\mbox{div}\left[(\epsilon+|\nabla v_\epsilon|^2)^{(p-2)/2}\nabla v_\epsilon\right]=0\mbox{ in }\Omega\\ v_\epsilon=g\mbox{ on }\partial\Omega\end{array}\right. \end{equation} Then, if the domain is regular "enough", we get that the solution $v_\epsilon$ is $C^{2,\rho}$ and it converges strongly to the solution of \begin{equation} (2)\qquad\left\lbrace \begin{array} \mbox{div}(|\nabla v|^{p-2}\nabla v) =0\mbox{ in }\Omega\\ v=g\mbox{ on }\partial\Omega\end{array}\right. \end{equation}

First, we can find this result in Gilbarg/Trudinger book, Chapter 15. Theorems 15.11, 15.18, 15.19.

In the book, this result works if the domain satisfies the exterior sphere condition. Therefore, the domain should be at least $C^2$ (I guess Ziemer's book about regularity says that it is enough $C^{1,\alpha}$).

(A) if we have a rectangular domain (in the plane), is this result still true? I mean, the regularity of solutions of (1) is still $C^{2,\rho}$?

(B) if (A) holds, then can we think about the problem in a $C^1$ by parts domain? For instance, one can think about the domain $$ \Omega=\lbrace 0<x<1, 0<y<h(x)\rbrace $$ where $h$ is a $1$-periodic, $C^1$ function.

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In the case of the $p$-harmonic equation on planar domains you can actually prove higher regularity: $v\in C^{k,\alpha}$, where $$ k+\alpha=\frac{1}{6}\left(7+\frac{1}{p-1}+\sqrt{1+\frac{14}{p-1}+\frac{1}{(p-1)^2}}\right). $$ While the formula seems insane, the result is actually sharp. This is the best possible estimate. You can find the proof in

T. Iwaniec, J. J. Manfredi, Regularity of $p$-harmonic functions on the plane. Rev. Mat. Iberoamericana 5 (1989), no. 1-2, 1–19.

I would recommend reading the paper since for the planar domains this is the right approach to the regularity. You can also check an earlier paper on the subject:

B. Bojarski, T. Iwaniec, $p$-harmonic equation and quasiregular mappings. Partial differential equations (Warsaw, 1984), 25–38, Banach Center Publ., 19, PWN, Warsaw, 1987.

This paper might be hard to find online, but if there is a need, I can try to find it.

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  • $\begingroup$ Yes, I saw that paper. But, the regularity of the p-harmonic function depends on p. For example, if We want $k=2$, then p have a smaller range of possible values, right? That is the reason I would like the approximation (1). $\endgroup$ – user143090 Oct 2 '19 at 17:23
  • $\begingroup$ @user143090 To get at least $k=2$ you need $p\leq 2$. Of course for $p=2$ you get $C^\infty$. $\endgroup$ – Piotr Hajlasz Oct 2 '19 at 19:17

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