On the geometry of the $p$-Laplacian operator

Abstract

The $p$-Laplacian operator ${\Delta }_{p}u=\mathrm{div}\left(|\nabla u{|}^{p-2}\nabla u\right)$ 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 $${\Delta }_p^{N}u:=\frac{1}{p}|\nabla u{|}^{2-p}{\Delta }_{p}u=\frac{1}{p}{\Delta }_1^{N}u+\frac{p-1}{p}{\Delta }_{\infty }^{N}u$$ 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.

15 pages, 5 figures, Survey lecture given at the WIAS conference "Theory and Applications of Partial Differential Equations" in Dec. 2015