Preprint 18/2022

A Harnack inequality for weak solutions of the Finsler γ-Laplacian

Max Goering

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Submission date: 06. May. 2022
Pages: 23
MSC-Numbers: 35J60, 35D30, 31C45
Keywords and phrases: Anisotropic, regularity, weak solutions
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Link to arXiv: See the arXiv entry of this preprint.

We study regularity of the Finsler γ-Laplacian, a general class of degenerate elliptic PDEs which naturally appear in anisotropic geometric problems. Precisely, given any strictly convex family of C1-norms {ρx} on n and γ > 1, we consider the W1(Ω) solutions of the anisotropic PDE

∫                            ∫    ⟨ρx(Du)γ−1(Dρx)(Du),D φ⟩ =   ⃗F ⋅Dφ + fφ    ∀φ ∈ W 1,γ′(Ω ).  Ω                            Ω                      0

Under the mild assumption |ξ|1ρx(ξ) [ν,Λ] for all (x,ξ) Ω × n and some 0 < ν Λ < we perform a Moser iteration, verifying that sub- and super-solutions satisfy one-sided ∥⋅∥ bounds, which together imply solutions are locally bounded. When u is non-negative this also implies a (weak) Harnack inequality. If f,⃗F 0 weak solutions also benefit from a strong maximum principle, and a Liouville-type theorem.

11.05.2022, 02:22