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

Junior da Silva Bessa, João Vitor da Silva, Laura Ospina

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.

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

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 where the operator 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.
Paper Structure (5 sections, 14 theorems, 127 equations)

This paper contains 5 sections, 14 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Schauder estimates: a brief mathematical tour
  3. Preliminaries
  4. Geometric Tangential Approach
  5. $C^{2, \text{Dini}}$ estimates: Proof of Theorem \ref{['MainThm01']}

Key Result

Theorem 1.1

Let $F: \text{Sym}(n) \to \mathbb{R}$ be a convex (or concave) uniformly elliptic operator satisfying $F(\mathbf{O}_n) = 0$ (where $\mathbf{O}_{n}$ denotes the $n \times n$ zero symmetric matrix). Let $u \in C^0(\mathrm{B}_1)$ be a viscosity solution of Then, for some $\alpha_{\mathrm{EK}} \in (0, 1)$ and constant $\mathrm{C} > 0$, depending only on the dimension $n$, and the ellipticity consta

Theorems & Definitions (26)

  • Theorem 1.1: Evans--Krylov's Theorem
  • Theorem 1.2: $C^{2,\alpha}$ regularity - dosPrazTei2016
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5: Local $C^{2,\textrm{Dini}}$ regularity
  • Remark 1.6
  • Corollary 1.7: $C^2$ implies $C^{2,\text{Dini}}$
  • Corollary 1.8
  • Theorem 1.9: Local $C^{2,\textrm{Dini}}$ regularity - Convex case
  • Remark 1.10
  • ...and 16 more