Search

MiS Preprint Repository

Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.

MiS Preprint
18/2022

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

Max Goering

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.

Received:
06.05.2022
Published:
09.05.2022
MSC Codes:
35J60, 35D30, 31C45
Keywords:
Anisotropic, regularity, weak solutions

Related publications

Preprint
2022 Repository Open Access
Max Goering

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