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Higher regularity of solutions to fully nonlinear elliptic equations

Thialita M. Nascimento, Ginaldo Sá, Aelson Sobral, Eduardo V. Teixeira

Abstract

We establish higher regularity properties of solutions to fully nonlinear elliptic equations at interior critical points. The key novelty of our estimates lies in the fact that they yield smoothness properties that go beyond the inherent regularity limitations dictated by the heterogeneity of the problem. We explore various scenarios, revealing a plethora of improved regularity estimates. Notably, depending on the model's parameters, we establish estimates that transcend the natural regularity regime of the model, from $C^{0,α_0}$ to $C^{1,α_1}$ and further to $C^{2,α_2}$, with the potential for even higher estimates.

Higher regularity of solutions to fully nonlinear elliptic equations

Abstract

We establish higher regularity properties of solutions to fully nonlinear elliptic equations at interior critical points. The key novelty of our estimates lies in the fact that they yield smoothness properties that go beyond the inherent regularity limitations dictated by the heterogeneity of the problem. We explore various scenarios, revealing a plethora of improved regularity estimates. Notably, depending on the model's parameters, we establish estimates that transcend the natural regularity regime of the model, from to and further to , with the potential for even higher estimates.
Paper Structure (18 sections, 28 theorems, 318 equations)

This paper contains 18 sections, 28 theorems, 318 equations.

Table of Contents

  1. Introduction
  2. Hypothesis and main results
  3. Preliminary definitions
  4. Assumptions and results
  5. Scaling properties
  6. Sharpness
  7. $C^{1,\alpha}$ regularity improvement
  8. Gain of regularity
  9. Asymptotic analysis
  10. $C^{2,\alpha}$ regularity improvement
  11. Hessian regularity at critical points
  12. Gain of regularity at inflection points
  13. Regularity below the dimension threshold
  14. Improved Hölder estimates
  15. Gradient continuity at vanishing points
  16. ...and 3 more sections

Key Result

Theorem 1

Let $u \in C (\overline{B_1} )$ be a normalized viscosity solution to Assume Assumptions unif ellipticity, convexity are in force and $F$ has a uniform continuous modulus of continuity in the coefficients. Then, $u$ is of class $C^{1,\epsilon_{m,\gamma,n,p}}$ at points in $\mathcal{C}(u)$, that is for all $x \in B_{\frac{1}{4}}(x_0)$, where $C>0$ is a universal constant, $x_0 \in \mathcal{C}(u)$

Theorems & Definitions (54)

  • Definition 1: Viscosity solution
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • ...and 44 more