Sharp regularity for degenerate obstacle type problems: a geometric approach

TL;DR

This paper establishes sharp regularity estimates for solutions of degenerate obstacle problems using a novel geometric approach, demonstrating $C^{1,eta}$ regularity and measure-zero free boundary under certain conditions.

Contribution

It introduces a new geometric method to prove sharp regularity for degenerate obstacle problems, including free boundary measure properties, which was not previously achieved.

Findings

Solutions are $C^{1,eta}$ regular with $eta= ext{min}igrace{ ext{H"older exponent of obstacle}, rac{1}{ ext{degeneracy}+1}igrace}$.

Free boundary has zero Lebesgue measure under non-degeneracy assumptions.

Method applies even to simple models like $|Du|^ ext{degeneracy} riangle u= ext{indicator}$.

Abstract

We prove sharp regularity estimates for solutions of obstacle type problems driven by a class of degenerate fully nonlinear operators; more specifically, we consider viscosity solutions of \[ |D u|^\gamma F(x, D^2u) = f(x)\chi_{\{u>\phi\}} \textrm{ in } B_1 \] with $\gamma>0$, $\phi \in C^{1, \alpha}(B_1)$ for some $\alpha\in(0,1]$ and $f\in L^\infty(B_1)$ constrained to satisfy \[ u\geq \phi\textrm{ in } B_1 \] and prove that they are $C^{1,\beta}(B_{1/2})$ (and in particular along free boundary points) where $\beta=\min\left\{\alpha, \frac{1}{\gamma+1}\right\}$. Moreover, we achieve such a feature by using a recently developed geometric approach which is a novelty for these kind of free boundary problems. Further, under a natural non-degeneracy assumption on the obstacle, we prove that the free boundary $\partial\{u>\phi\}$ has zero Lebesgue measure. Our results are new even for seemingly simple model as follows \[ |Du|^\gamma \Delta u=\chi_{\{u>\phi\}} \quad \text{with}\quad \gamma>0. \]