Hölder gradient estimates for a class of singular or degenerate parabolic equations

Advances in Nonlinear Analysis

article

Cyril Imbert¹  Tianling Jin²  Luis Silvestre³ 

[1] Department of Mathematics and Applications;Department of Mathematics, The Hong Kong University of Science and Technology;Department of Mathematics, The University of Chicago, 5734 S. University Avenue

关键词: Regularity;    degenerate parabolic equations;    singular parabolic equations;   

DOI  :  10.1515/anona-2016-0197

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

We prove interior Hölder estimates for the spatial gradients of the viscosity solutions to the singular or degenerate parabolic equation u t = | ∇ ⁡ u | κ ⁢ div ⁡ ( | ∇ ⁡ u | p - 2 ⁢ ∇ ⁡ u ) , u_{t}=\lvert\nabla u\rvert^{\kappa}\operatorname{div}(\lvert\nabla u\rvert^{p-% 2}\nabla u), where p ∈ ( 1 , ∞ ) {p\in(1,\infty)} and κ ∈ ( 1 - p , ∞ ) {\kappa\in(1-p,\infty)} . This includes the from L ∞ {L^{\infty}} to C 1 , α {C^{1,\alpha}} regularity for parabolic p -Laplacian equations in both divergence form with κ = 0 {\kappa=0} , and non-divergence form with κ = 2 - p {\kappa=2-p} .

【 授权许可】

CC BY