Higher order interpolative geometries and gradient regularity in evolutionary obstacle problems

2023/08/30 by Sunghan Kim, Kim, Sunghan, Kaj Nyström +1
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Geometric Analysis and Curvature Flows #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.2308.15818

Abstract

We prove new optimal C^1,α regularity results for obstacle problems involving evolutionary p-Laplace type operators in the degenerate regime p > 2. Our main results include the optimal regularity improvement at free boundary points in intrinsic backward p-paraboloids, up to the critical exponent, α≤ 2/(p-2), and the optimal regularity across the free boundaries in the full cylinders up to a universal threshold. Moreover, we provide an intrinsic criterion by which the optimal regularity improvement at free boundaries can be extended to the entire cylinders. An important feature of our analysis is that we do not impose any assumption on the time derivative of the obstacle. Our results are formulated in function spaces associated to what we refer to as higher order or C^1,α intrinsic interpolative geometries.

Related

• Optimal regularity for the obstacle problem for the p-Laplacian
• Regularity theory for nonlocal obstacle problems with critical and subcritical scaling
• Optimal regularity for supercritical parabolic obstacle problems
• Irregular Time Dependent Obstacles
• Global gradient estimates for nonlinear obstacle problems with nonstandard growth