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

Max Goering

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Submission date: 06. May. 2022
Pages: 23
Bibtex
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.

Abstract:
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 {ρ_x} on ℝⁿ and γ > 1, we consider the W^1,γ(Ω) solutions of the anisotropic PDE

Under the mild assumption |ξ|⁻¹ρ_x(ξ) ∈ [ν,Λ] for all (x,ξ) ∈ Ω × ℝⁿ 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, ≡ 0 weak solutions also benefit from a strong maximum principle, and a Liouville-type theorem.