Let $\Omega \subset\mathbb{R}^d$, $d \geq 2$, be a sufficiently nice set to make the following question meaningful.

I am interested in the space of p-harmonic functions on $\Omega$; that is, the metric space $$P = \{ u \in W^{1,p}(\Omega); \Delta_p u = 0 \}.$$ I don't impose any boundary values, so the space is not a singleton.

Has this space been investigated? Is it, for example, a manifold, or does it have some other structure?

If $p=2$, the set $P$ is a linear space, since the Laplacian is linear. Keywords or references even in the linear case would be useful.

This question is inspired by Joonas' question Density of restrictions of $p$-harmonic functions on a hypersurface .


IN $d=2$ p-harmonic functions are more regular the the usual normal $C^{1,\alpha}$ regularity (see Iwaniec-Manfredi, Rev Mat Iberoamericana 1989) and have additional nice geometric properties such as unique continuation (still unknown in higher dimension) and K-quasiconformality (see the nice paper by Manfredi, Proc AMS, 1988). This should help getting more info about the structure of $P$ you look for at least in the plane.


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