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

TL;DR

This paper establishes a Harnack inequality and regularity results for weak solutions of the anisotropic Finsler gamma-Laplacian, a class of degenerate elliptic PDEs arising in geometric problems, under mild assumptions.

Contribution

It introduces new regularity and boundedness results for solutions of the Finsler gamma-Laplacian, including Harnack inequality, maximum principle, and Liouville theorem, using Moser iteration.

Findings

Solutions are locally bounded under mild conditions.

Non-negative solutions satisfy a weak Harnack inequality.

Weak solutions obey a strong maximum principle and Liouville theorem.

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 $$ \displaystyle \int_{\Omega} \left \langle \rho_{x}(Du)^{\gamma-1} (D \rho_{x})(Du), D \varphi \right \rangle = \int_{\Omega} \vec{F} \cdot D \varphi + f \varphi \qquad \forall \varphi \in W^{1,\gamma^{\prime}}_{0}(\Omega). $$ 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 $\| \cdot \|_{\infty}$ 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.