On the geometry of the $p$-Laplacian operator

TL;DR

This paper explores the geometric aspects of the $p$-Laplacian operator, its eigenfunctions, and limits as $p$ approaches 1, 2, or infinity, including the normalized and game-theoretic variants.

Contribution

It reviews known results on eigenfunctions of the classical 2-Laplacian and extends these to general $p$, also discussing properties of the normalized $p$-Laplacian and its parabolic form.

Findings

Eigenfunctions relate to the geometry of domain $\\Omega$ for $p$ near 1 or infinity.

Normalized $p$-Laplacian $\\Delta_p^N$ is uniformly elliptic for all $p \in (1,\infty)$.

The normalized $p$-Laplacian is homogeneous of degree 1 and not of divergence type.

Abstract

The $p$-Laplacian operator $\Delta_pu={\rm div }\left(|\nabla u|^{p-2}\nabla u\right)$ is not uniformly elliptic for any $p\in(1,2)\cup(2,\infty)$ and degenerates even more when $p\to \infty$ or $p\to 1$. In those two cases the Dirichlet and eigenvalue problems associated with the $p$-Laplacian lead to intriguing geometric questions, because their limits for $p\to\infty$ or $p\to 1$ can be characterized by the geometry of $\Omega$. In this little survey we recall some well-known results on eigenfunctions of the classical 2-Laplacian and elaborate on their extensions to general $p\in[1,\infty]$. We report also on results concerning the normalized or game-theoretic $p$-Laplacian $$\Delta_p^Nu:=\tfrac{1}{p}|\nabla u|^{2-p}\Delta_pu=\tfrac{1}{p}\Delta_1^Nu+\tfrac{p-1}{p}\Delta_\infty^Nu$$ and its parabolic counterpart $u_t-\Delta_p^N u=0$. These equations are homogeneous of degree 1 and $\Delta_p^N$ is uniformly elliptic for any $p\in (1,\infty)$. In this respect it is more benign than the $p$-Laplacian, but it is not of divergence type.