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Interior regularity estimates for fully nonlinear equations with arbitrary nonhomogeneous degeneracy laws

Pêdra D. S. Andrade, Thialita M. Nascimento

TL;DR

The paper studies interior regularity for viscosity solutions of fully nonlinear degenerate elliptic equations with arbitrary nonhomogeneous degeneracy laws, establishing local differentiability under precise conditions on the degeneracy moduli and using a novel improvement-of-flatness framework. The authors blend a renormalization/approximation approach that connects the degenerate model to the homogeneous equation $F(D^2u)=0$ with a noncollapsing degeneracy analysis via shored-up moduli, built on Crandall–Ishii–Lions theory. They prove universal Hölder continuity for the nonhomogeneous model and then achieve $C^{1,eta}$ regularity through an iterative affine-approximation scheme, producing a modulus of continuity for the gradient and extending to multi-phase degeneracies. The results advance the regularity theory for degenerate, nonhomogeneous fully nonlinear PDEs and have potential applications in materials science and related variational problems with nonstandard growth.

Abstract

In this paper, we study regularity estimates for a class of degenerate, fully nonlinear elliptic equations with arbitrary nonhomogeneous degeneracy laws. We establish that viscosity solutions are locally continuously differentiable under suitable conditions on the degeneracy laws. Our proof employs improvement of flatness techniques alongside an alternative recursive algorithm for renormalizing the approximating solutions, linking our model to the homogeneous, fully nonlinear, uniformly elliptic equation.

Interior regularity estimates for fully nonlinear equations with arbitrary nonhomogeneous degeneracy laws

TL;DR

The paper studies interior regularity for viscosity solutions of fully nonlinear degenerate elliptic equations with arbitrary nonhomogeneous degeneracy laws, establishing local differentiability under precise conditions on the degeneracy moduli and using a novel improvement-of-flatness framework. The authors blend a renormalization/approximation approach that connects the degenerate model to the homogeneous equation with a noncollapsing degeneracy analysis via shored-up moduli, built on Crandall–Ishii–Lions theory. They prove universal Hölder continuity for the nonhomogeneous model and then achieve regularity through an iterative affine-approximation scheme, producing a modulus of continuity for the gradient and extending to multi-phase degeneracies. The results advance the regularity theory for degenerate, nonhomogeneous fully nonlinear PDEs and have potential applications in materials science and related variational problems with nonstandard growth.

Abstract

In this paper, we study regularity estimates for a class of degenerate, fully nonlinear elliptic equations with arbitrary nonhomogeneous degeneracy laws. We establish that viscosity solutions are locally continuously differentiable under suitable conditions on the degeneracy laws. Our proof employs improvement of flatness techniques alongside an alternative recursive algorithm for renormalizing the approximating solutions, linking our model to the homogeneous, fully nonlinear, uniformly elliptic equation.
Paper Structure (8 sections, 8 theorems, 169 equations)

This paper contains 8 sections, 8 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main assumptions and definitions
  4. Foundational results
  5. Compactness for a family of PDEs with arbitrary nonhomogeneous degeneracy
  6. Improvement of flatness
  7. Regularity results
  8. The construction of the shored-up sequence

Key Result

Theorem 1.1

Let $u \in \mathcal{C}(B_1)$ be a normalized viscosity solution to where $0\le a(\cdot) \in \mathcal{C}(B_1)$ and $\sigma(\cdot), \nu(\cdot)$ are moduli of continuity with inverses $\sigma^{-1}$, $\nu^{-1}$. Suppose $F$ is a fully nonlinear $(\lambda, \Lambda)$-uniformly elliptic operator, $\nu (t) = o(\sigma(t))$, $\nu^{-1}$ is Dini continuous, and $f \in L^{\inft for every $x, y \in B_{1/4}$.

Theorems & Definitions (27)

  • Theorem 1.1: Differentiability of the solutions
  • Definition 2.1: Pucci's extremal operators
  • Definition 2.2: Dini condition
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5: APPT
  • Remark 2.6
  • Definition 2.7: APPT, Shoring-up
  • Definition 2.8: Non-collapsing property
  • Proposition 2.9
  • ...and 17 more