Geometric gradient estimates for fully nonlinear models with non-homogeneous degeneracy and applications

Abstract

We establish sharp $$C_{\text {loc}}^{1, \beta }$$ geometric regularity estimates for bounded solutions of a class of fully nonlinear elliptic equations with non-homogeneous degeneracy, whose model equation is given by $$\begin{aligned} \left[ |Du|^p+\mathfrak {a}(x)|Du|^q\right] {\mathcal {M}}_{\lambda , \Lambda }^{+}(D^2 u)= f(x, u) \quad \text {in} \quad \Omega , \end{aligned}$$ for a bounded and open set $$\Omega \subset {\mathbb {R}}^N$$ , and appropriate data $$p, q \in (0, \infty )$$ , $$\mathfrak {a}$$ and f. Such regularity estimates simplify and generalize, to some extent, earlier ones via totally different modus operandi. Our approach is based on geometric tangential methods and makes use of a refined oscillation mechanism combined with compactness and scaling techniques. In the end, we present some connections of our findings with a variety of nonlinear geometric free boundary problems and relevant nonlinear models in the theory of elliptic PDEs, which may have their own interest. We also deliver explicit examples where our results are sharp.