Schauder estimates for flat solutions to a class of fully nonlinear elliptic PDEs with Dini continuous data: a geometric tangential approach

TL;DR

This paper establishes local Schauder estimates for flat viscosity solutions to a broad class of fully nonlinear elliptic PDEs with Dini continuous data, using geometric tangential techniques and perturbative arguments.

Contribution

It introduces a novel geometric tangential approach to derive Schauder estimates for solutions with Dini continuity, extending previous results to non-convex operators with linear drift.

Findings

Established Schauder estimates for flat solutions with Dini data

Derived an Evans-Krylov type regularity estimate

Characterized nodal sets of solutions in the nonlinear elliptic framework

Abstract

In this manuscript, we establish local Schauder estimates for flat viscosity solutions, that is, solutions with sufficiently small norms, to a class of fully nonlinear elliptic partial differential equations of the form \[ F(D^{2} u, x) + \langle \mathfrak{B}(x), D u \rangle = f(x) \quad \text{in} \quad \mathrm{B}_1 \subset \mathbb{R}^{n}, \] where the operator \(F\) is differentiable, though not necessarily convex or concave. In addition, we impose suitable Dini-type continuity assumptions on the data. Our methodology is based on geometric tangential techniques, combined with compactness and perturbative arguments. This approach is strongly motivated by recent advances in the theory of nonlinear elliptic equations and free boundary problems. As a byproduct of our analysis, we also obtain an Evans-Krylov type estimate. Our results can be viewed as an extension of the work by dos Prazeres and Teixeira (Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 15 (2016), 485-500), now within the framework of linear drift terms and Dini continuity assumptions. Finally, we apply our results to characterize the nodal sets of flat viscosity solutions of non-convex, fully nonlinear, uniformly elliptic PDEs.