论文标题
Finsler $γ$ -laplacian的弱解决方案的harnack不平等
A Harnack inequality for weak solutions of the Finsler $γ$-Laplacian
论文作者
论文摘要
我们研究了Finsler $γ$ -laplacian的规律性，这是一种普通的退化椭圆形PDE，自然出现在各向异性几何问题中。确切地说，给定$ c^{1} $的任何严格凸族家族$ \ {ρ_{ρ_{x} \} $ on $ \ mathbb {r}^{n}^{n} $和$γ> 1 $ \langleρ_{x}（du）^{γ-1}（dρ_{x}）（du），dφ\ right \ rangle = \int_Ω w^{1，γ^{\ prime}} _ {0}（ω）。 $$在温和的假设下$ | em | e |^{ - 1}ρ_{x}（ξ）（ξ）\ in [ν，λ] $ in [ν，λ] $ for所有$（x，ch，ξ）\ inω\ times \ times \ times \ times \ mathbb {r}^{n} $ $ \ | \ cdot \ | _ {\ infty} $ bounds，这意味着解决方案是本地边界的。当$ u $不负时，这也意味着（弱）harnack不平等。如果$ f，\ vec {f} \ equiv 0 $弱解决方案也受益于强大的最大原则和liouville-type定理。
We study regularity of the Finsler $γ$-Laplacian, a general class of degenerate elliptic PDEs which naturally appear in anisotropic geometric problems. Precisely, given any strictly convex family of $C^{1}$-norms $\{ ρ_{x}\}$ on $\mathbb{R}^{n}$ and $γ> 1$, we consider the $W^{1,γ}(Ω)$ solutions of the anisotropic PDE $$ \displaystyle \int_Ω \left \langle ρ_{x}(Du)^{γ-1} (D ρ_{x})(Du), D φ\right \rangle = \int_Ω \vec{F} \cdot D φ+ f φ\qquad \forall φ\in W^{1,γ^{\prime}}_{0}(Ω). $$ Under the mild assumption $|ξ|^{-1} ρ_{x}( ξ) \in [ν, Λ]$ for all $(x,ξ) \in Ω\times \mathbb{R}^{n}$ and some $0 < ν\le Λ< \infty$ 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.
