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Schauder-type estimates for fully nonlinear degenerate elliptic equations

Thialita M. Nascimento

Abstract

In this paper, we examine regularity estimates for solutions to fully nonlinear, degenerated elliptic equations, at interior vanishing source points. At these points, we obtain Schauder-type regularity estimates, which depend on the Hölder-like source-ellipticity vanishing rate.

Schauder-type estimates for fully nonlinear degenerate elliptic equations

Abstract

In this paper, we examine regularity estimates for solutions to fully nonlinear, degenerated elliptic equations, at interior vanishing source points. At these points, we obtain Schauder-type regularity estimates, which depend on the Hölder-like source-ellipticity vanishing rate.
Paper Structure (9 sections, 8 theorems, 132 equations)

This paper contains 9 sections, 8 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Assumptions, main results and further insights
  3. Improved gradient regularity
  4. Gradient Approximations
  5. Regularity for small gradient
  6. Sharp regularity
  7. Hessian continuity at local extrema
  8. Flatness estimate
  9. Higher regularity

Key Result

Theorem 1

Let $u \in C(B_1)$ be a bounded viscosity solution to main eq intr. Assume that $F:B_1 \times Sym(n) \to \mathbb{R}$ satisfy unif ellipticity, contin of coeff, and that $f$ satisfies holder cont of f. Then $u$ is of class $C^{1, \min\{\alpha_0^{-}, \frac{1+\theta}{1 + \gamma}\} }$ at the origin. Tha we have , for all $x \in B_{1/4}(0)$, where $C_{\beta} > 0$ depending only on $\beta$, and univers

Theorems & Definitions (14)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Lemma 1: Approximation lemma
  • proof
  • Lemma 2: Discrete regularity
  • proof
  • Proposition 1
  • proof
  • ...and 4 more