A Harnack inequality for weak solutions of the Finsler $\gamma$-Laplacian

Abstract

We study regularity of the Finsler $\gamma$-Laplacian, a general class of degenerate elliptic PDEs which naturally appear in anisotropic geometric problems. Precisely, given any strictly convex family of $C^1$-norms $\{{\rho }_x\}$ on ${\mathbb{R}}^n$ and $\gamma >1$, we consider the $W^{1,\gamma }(\Omega )$ solutions of the anisotropic PDE $${\int }_{\Omega }⟨{\rho }_x(Du{)}^{\gamma -1}(D{\rho }_x)(Du),D\phi ⟩={\int }_{\Omega }\vec{F}\cdot D\phi +f\phi \forall \phi \in W_0^{1,{\gamma }^{\prime }}(\Omega ).$$ Under the mild assumption $|\xi {|}^{-1}{\rho }_x(\xi )\in [\nu ,\Lambda ]$ for all $(x,\xi )\in \Omega \times {\mathbb{R}}^n$ and some $0<\nu \le \Lambda <\infty$ we perform a Moser iteration, verifying that sub- and super-solutions satisfy one-sided $\Vert \cdot {\Vert }_{\infty }$ bounds, which together imply solutions are locally bounded. When $u$ is non-negative this also implies a (weak) Harnack inequality. If $f,\vec{F}\equiv 0$ weak solutions also benefit from a strong maximum principle, and a Liouville-type theorem.

arXiv commentNew introduction. Strengthened Liouville Theorem (Theorem 3.9) added Bernstein Theorem (Theorem 1.5) and Remark 1.6 is new