Higher order interpolative geometries and gradient regularity in evolutionary obstacle problems

TL;DR

This paper establishes new optimal regularity results for evolutionary obstacle problems involving degenerate p-Laplace operators, focusing on free boundary behavior and intrinsic geometries without assuming obstacle time derivatives.

Contribution

It introduces novel regularity improvements at free boundaries in intrinsic geometries for p > 2, extending regularity results without obstacle time derivative assumptions.

Findings

Optimal $C^{1,eta}$ regularity at free boundary points

Regularity extension criteria for entire cylinders

No assumptions on obstacle time derivatives

Abstract

We prove new optimal $C^{1,\alpha}$ 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, $\alpha \leq 2/(p-2)$, 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 $C^{1,\alpha}$ intrinsic interpolative geometries.