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MiS Preprint
18/2022
A Harnack inequality for weak solutions of the Finsler $\gamma$-Laplacian
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.
