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Regularity in diffusion models with gradient activation

Damião Araújo, Aelson Sobral, Eduardo V. Teixeira

TL;DR

This work analyzes gradient-activated diffusion modeled by $(|Du|-\\kappa)_+^q F(D^2u)=f$, producing a sharp regularity theory for highly degenerate fully nonlinear elliptic equations with an unknown free boundary. By introducing $\kappa$-grad viscosity solutions and a De Giorgi–type gradient-oscillation improvement, the authors prove a universal Lipschitz bound independent of the degeneracy $q$ and establish $C^1$ regularity up to the free boundary via a gradient-level modulus of continuity. The methods combine compactness under scaling, localized near-boundary oscillation control, and far-field elliptic regularity, yielding results that extend to unconstrained free boundary problems, infinite degeneracy limits, and flame-propagation models with obstacles. The findings provide robust gradient control and boundary regularity for gradient-driven diffusion, with broad implications for superconductivity, random surfaces, traffic flow, and combustion-type phenomena.

Abstract

We prove sharp regularity estimates for solutions of highly degenerate fully nonlinear elliptic equations. These are free boundary models in which a nonlinear diffusion process drives the system only in the region where the gradient surpasses a given threshold. Our main result concerns the existence of a universal modulus of continuity for $Du$, up to the free boundary. Gradient bounds for the $L^\infty$ norm are proven to be uniform with respect to the degree of degeneracy. Several new ingredients are needed and among the tools introduced in this paper is an improvement of regularity lemma designed to measure the oscillation decay concerning the gradient level-set distance. Applications of the methods are discussed at the end of the paper.

Regularity in diffusion models with gradient activation

TL;DR

This work analyzes gradient-activated diffusion modeled by , producing a sharp regularity theory for highly degenerate fully nonlinear elliptic equations with an unknown free boundary. By introducing -grad viscosity solutions and a De Giorgi–type gradient-oscillation improvement, the authors prove a universal Lipschitz bound independent of the degeneracy and establish regularity up to the free boundary via a gradient-level modulus of continuity. The methods combine compactness under scaling, localized near-boundary oscillation control, and far-field elliptic regularity, yielding results that extend to unconstrained free boundary problems, infinite degeneracy limits, and flame-propagation models with obstacles. The findings provide robust gradient control and boundary regularity for gradient-driven diffusion, with broad implications for superconductivity, random surfaces, traffic flow, and combustion-type phenomena.

Abstract

We prove sharp regularity estimates for solutions of highly degenerate fully nonlinear elliptic equations. These are free boundary models in which a nonlinear diffusion process drives the system only in the region where the gradient surpasses a given threshold. Our main result concerns the existence of a universal modulus of continuity for , up to the free boundary. Gradient bounds for the norm are proven to be uniform with respect to the degree of degeneracy. Several new ingredients are needed and among the tools introduced in this paper is an improvement of regularity lemma designed to measure the oscillation decay concerning the gradient level-set distance. Applications of the methods are discussed at the end of the paper.
Paper Structure (13 sections, 15 theorems, 185 equations, 1 figure)

This paper contains 13 sections, 15 theorems, 185 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Uniform Lipschitz estimates
  4. Compactness for scaled PDEs
  5. $C^1$ regularity up to the free boundary
  6. Improvement of oscillation near the free boundary
  7. Regularity estimates far from the free boundary
  8. Proof of Proposition \ref{['unif_c1_thm']}
  9. Concluding the proof of Theorem \ref{['C1THM']}
  10. Applications
  11. Unconstrained free boundary problems
  12. PDE models with infinite degree of degeneracy
  13. Flame propagation with an obstacle

Key Result

Theorem 3.1

Let $u$ be a $\kappa$-grad viscosity solution of maineq in $B_1$. Then $u$ is Lipschitz continuous in $B_{1/2}$, with universal bounds. More precisely, there exists a constant $C$ depending only on $n$, $\lambda$, $\Lambda$, $\kappa$, $\|f\|_\infty$ and $\|u\|_\infty$, but not on $q$, such that

Figures (1)

  • Figure 1: This figure is a representation the geometry of the problem. The white region, $\{ |Du| > \kappa\}$, displays the part of the domain in which a diffusion PDE drives the system. In the dark grey zone, $\{ |Du| \le \kappa\}$, the system is dormant. The analysis in the intermediary light grey sector, $\{\kappa < |Du| < \kappa + \mu \}$, for some $0< \mu \ll 1$, is critical for the proof of Theorem \ref{['C1THM']}. It is worth highlighting, however, that the topology of such a regions can be much more complicated and their corresponding boundaries highly irregulars. This is why Theorem \ref{['C1THM']} is a non-trivial (somewhat striking) result.

Theorems & Definitions (31)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.1
  • Theorem 3.1
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['LipTHM']}
  • Lemma 4.1
  • ...and 21 more