Sharp regularity for degenerate obstacle type problems: a geometric approach

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 \[ |D u|^γF(x, D^2u) = f(x)χ_{\{u>ϕ\}} \textrm{ in } B_1 \] with $\gamma >0$, $\varphi \in C^{1,\alpha }(B_1)$ for some $\alpha \in (0,1]$ and $f\in L^{\infty }(B_1)$ constrained to satisfy \[ u\geq ϕ\textrm{ in } B_1 \] 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 $\partial \{u>\varphi \}$ has zero Lebesgue measure. Our results are new even for seemingly simple model as follows \[ |Du|^γΔu=χ_{\{u>ϕ\}} \quad \text{with}\quad γ>0. \]