Universal regularity estimates for solutions to fully nonlinear elliptic equations with oblique boundary data

Abstract: In this work, we establish universal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations with oblique boundary conditions, whose general model is given by $$ \left{ \begin{array}{rcl} F(D^2u,x) &=& f(x) \quad \mbox{in} \,\, \Omega\ \beta(x) \cdot Du(x) + \gamma(x) \, u(x)&=& g(x) \quad \mbox{on} \,\, \partial \Omega. \end{array} \right. $$ Such regularity estimates are achieved by exploring the integrability properties of $f$ based on different scenarios, making a $\text{VMO}$ assumption on the coefficients of $F$, and by considering suitable smoothness properties on the boundary data $\beta, \gamma$ and $g$. Particularly, we derive sharp estimates for borderline cases where $f \in L^n(\Omega)$ and $f\in p-\textrm{BMO}(\Omega)$. Additionally, for source terms in $L^p(\Omega)$, for $p \in (n, \infty)$, we obtain sharp gradient estimates. Finally, we also address Schauder-type estimates for convex/concave operators and suitable H\"{o}lder data.