On the geometry of the p -Laplacian operator

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