Sharp regularity for degenerate obstacle type problems: a geometric approach

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 $γ>0$, $ϕ\in C^{1, α}(B_1)$ for some $α\in(0,1]$ and $f\in L^\infty(B_1)$ constrained to satisfy $u\geq ϕ\textrm{ in } B_1$ and prove that they are $C^{1,β}(B_{1/2})$ (and in particular along free boundary points) where $β=\min\left\{α, \frac{1}{γ+1}\right\}$. 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 $\partial\{u>ϕ\}$ has zero Lebesgue measure. Our results are new even for seemingly simple model as follows $|Du|^γΔu=χ_{\{u>ϕ\}} \quad \text{with}\quad γ>0.$

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