# regularity of p-harmonic functions

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) $$$$(1)\qquad \left\lbrace \begin{array}\mbox{div}\left[(\epsilon+|\nabla v_\epsilon|^2)^{(p-2)/2}\nabla v_\epsilon\right]=0\mbox{ in }\Omega\\ v_\epsilon=g\mbox{ on }\partial\Omega\end{array}\right.$$$$ Then, if the domain is regular "enough", we get that the solution $$v_\epsilon$$ is $$C^{2,\rho}$$ and it converges strongly to the solution of $$$$(2)\qquad\left\lbrace \begin{array} \mbox{div}(|\nabla v|^{p-2}\nabla v) =0\mbox{ in }\Omega\\ v=g\mbox{ on }\partial\Omega\end{array}\right.$$$$

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 $$\Omega=\lbrace 0 where $$h$$ is a $$1$$-periodic, $$C^1$$ function.

• Where did you see the method? Apr 19, 2023 at 19:14

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.
B. Bojarski, T. Iwaniec, $$p$$-harmonic equation and quasiregular mappings. Partial differential equations (Warsaw, 1984), 25–38, Banach Center Publ., 19, PWN, Warsaw, 1987.
• 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). Oct 2, 2019 at 17:23
• @user143090 To get at least $k=2$ you need $p\leq 2$. Of course for $p=2$ you get $C^\infty$. Oct 2, 2019 at 19:17