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- W4287688001 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." @default.
- W4287688001 created "2022-07-26" @default.
- W4287688001 creator A5044330906 @default.
- W4287688001 date "2020-08-18" @default.
- W4287688001 modified "2023-09-25" @default.
- W4287688001 title "A Harnack inequality for weak solutions of the Finsler $gamma$-Laplacian" @default.
- W4287688001 doi "https://doi.org/10.48550/arxiv.2008.08028" @default.
- W4287688001 hasPublicationYear "2020" @default.
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