  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.
