Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A quantitative Hopf-Oleinik lemma for degenerate fully nonlinear operators and applications to free boundary problems

Davide Giovagnoli, Enzo Maria Merlino, Diego Moreira

TL;DR

The paper develops a quantitative Hopf-Oleinik framework for degenerate fully nonlinear elliptic equations of the form $|\nabla u|^{\alpha}F(D^{2}u)=f$, yielding linear boundary growth and $L^{\varepsilon}$ interior control with universal constants. It combines barrier constructions with large-gradient Harnack theory to obtain a robust set of estimates, including a two-sided Pucci comparison and a counterexample showing the necessity of the full operator regime. The authors extend these PDE tools to free boundary problems, proving Lipschitz regularity for the one-phase Bernoulli problem and ε-uniform Lipschitz bounds for a flame-propagation model, and they provide a Sobolev gluing result across rough interfaces to handle interfaces in these problems. Collectively, the results advance quantitative boundary nondegeneracy and gradient control for degenerate fully nonlinear operators, with direct implications for free boundary regularity and combustion-type models.

Abstract

We prove a quantitative inhomogeneous Hopf-Oleinik lemma for viscosity solutions of $$|\nabla u|^αF(D^{2}u)=f $$ and, more generally, for viscosity supersolutions of $|\nabla u|^α\,{M}^-_{λ,Λ}(D^{2}u)\le f$. The result yields linear boundary growth with universal constants depending only on the structural data. We also exhibit a counterexample showing that the Hopf lemma fails for equations that act only in the large-gradient regime (in the sense of Imbert and Silvestre), thereby delineating the scope of our theorem. As applications, we obtain Lipschitz regularity for viscosity solutions of one-phase Bernoulli free boundary problems driven by these degenerate fully nonlinear operators and derive $\varepsilon$-uniform Lipschitz bounds for a one-phase flame propagation model.

A quantitative Hopf-Oleinik lemma for degenerate fully nonlinear operators and applications to free boundary problems

TL;DR

The paper develops a quantitative Hopf-Oleinik framework for degenerate fully nonlinear elliptic equations of the form , yielding linear boundary growth and interior control with universal constants. It combines barrier constructions with large-gradient Harnack theory to obtain a robust set of estimates, including a two-sided Pucci comparison and a counterexample showing the necessity of the full operator regime. The authors extend these PDE tools to free boundary problems, proving Lipschitz regularity for the one-phase Bernoulli problem and ε-uniform Lipschitz bounds for a flame-propagation model, and they provide a Sobolev gluing result across rough interfaces to handle interfaces in these problems. Collectively, the results advance quantitative boundary nondegeneracy and gradient control for degenerate fully nonlinear operators, with direct implications for free boundary regularity and combustion-type models.

Abstract

We prove a quantitative inhomogeneous Hopf-Oleinik lemma for viscosity solutions of and, more generally, for viscosity supersolutions of . The result yields linear boundary growth with universal constants depending only on the structural data. We also exhibit a counterexample showing that the Hopf lemma fails for equations that act only in the large-gradient regime (in the sense of Imbert and Silvestre), thereby delineating the scope of our theorem. As applications, we obtain Lipschitz regularity for viscosity solutions of one-phase Bernoulli free boundary problems driven by these degenerate fully nonlinear operators and derive -uniform Lipschitz bounds for a one-phase flame propagation model.

Paper Structure

This paper contains 14 sections, 14 theorems, 288 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Notation and preliminaries
  4. Construction of a barrier with geometry
  5. Weak Harnack and Harnack Inequalities
  6. Weak Harnack Inequality
  7. Harnack Inequality
  8. Proof of the quantitative version of the Hopf Lemma
  9. Failure of the Hopf Lemma in the large gradient regime
  10. Gradient estimates for solutions to $|\nabla u|^{\alpha} F(D^2 u)=f$
  11. Gluing Sobolev functions across a rough interface
  12. Lipschitz regularity of the one-phase Bernoulli problem
  13. Uniform Lipschitz continuity for a flame propagation model
  14. From Campanato to $C^{1,\gamma}$ Regularity

Key Result

Theorem 2.1

Let $\alpha\ge0$ and $0<\lambda\le\Lambda$. Let $u\in C(\overline{B_1})$ be a viscosity supersolution of where $f\in C(B_1)\cap L^\infty(B_1)$ and $u\ge0$ in $B_1$. Then there exist constants $\varepsilon\in(0,1)$ and $A_1,A_2,C>0$, depending only on $n,\lambda,\Lambda,\alpha$, such that:

Theorems & Definitions (41)

  • Theorem 2.1: Quantitative Hopf-Oleinik lemma
  • Proposition 2.2: Failure of the Hopf lemma in the large gradient regime
  • Theorem 2.3: Lipschitz regularity for a one-phase Bernoulli problem
  • Theorem 2.4: Lipschitz regularity for the flame propagation problem
  • Remark 3.1
  • Definition 3.2
  • Remark 3.3
  • Remark 3.4: Rescaling while preserving ellipticity
  • Definition 3.5
  • Definition 3.6
  • ...and 31 more