On the geometry of the $p$-Laplacian operator

The $p$-Laplacian operator $Δ_pu={\rm div }\left(|\nabla u|^{p-2}\nabla u\right)$ 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 $Ω$. 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 $Δ_p^Nu:=\tfrac{1}{p}|\nabla u|^{2-p}Δ_pu=\tfrac{1}{p}Δ_1^Nu+\tfrac{p-1}{p}Δ_\infty^Nu$ and its parabolic counterpart $u_t-Δ_p^N u=0$. These equations are homogeneous of degree 1 and $Δ_p^N$ 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.

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