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Some Comparison Results for First-Order Hamilton-Jacobi Equations and Second-Order Fully Nonlinear Parabolic Equations with Ventcell Boundary Conditions

Guy Barles, Emmanuel Chasseigne

TL;DR

The paper establishes the first global comparison principle for Ventcell boundary conditions in viscosity-solution settings for fully nonlinear parabolic equations. It develops a localized flat-boundary analysis via boundary flattening and introduces a twin-blow-up scaling adapted to Ventcell BC to prove local comparison results in the half-space, under normal coercivity (first-order) or normal strong ellipticity (second-order) hypotheses and quasiconvexity of the boundary operator. By reducing global results to local ones, the authors obtain (GCR) in general smooth domains through localization, and they provide auxiliary tools such as sup-convolution regularization and convexity-preserving subsolution constructions. The results enable Perron-based existence and give insight into boundary behavior and regularization, with open questions on regularity and large-time behavior.

Abstract

In this article, we consider fully nonlinear, possibly degenerate, parabolic equations associated with Ventcell boundary conditions in bounded or unbounded, smooth domains. We first analyze the exact form of such boundary conditions in general domains in order that the notion of viscosity solutions makes sense. Then we prove general comparison results, both for first- and second-order equations, under rather natural assumptions on the nonlinearities: $(i)$ in the second-order case, the only restrictive assumption is that the equation has to be strictly elliptic in the normal direction, in a neighborhood of the boundary; $(ii)$ in the first-order one, quasiconvexity assumptions have to be imposed both on the equation and the boundary condition, the equation being coercive in the normal direction. Our method is inspired by the ``twin blow-up method'' of Forcadel-Imbert-Monneau, that we adapt to a scaling consistent with the Ventcell boundary condition.

Some Comparison Results for First-Order Hamilton-Jacobi Equations and Second-Order Fully Nonlinear Parabolic Equations with Ventcell Boundary Conditions

TL;DR

The paper establishes the first global comparison principle for Ventcell boundary conditions in viscosity-solution settings for fully nonlinear parabolic equations. It develops a localized flat-boundary analysis via boundary flattening and introduces a twin-blow-up scaling adapted to Ventcell BC to prove local comparison results in the half-space, under normal coercivity (first-order) or normal strong ellipticity (second-order) hypotheses and quasiconvexity of the boundary operator. By reducing global results to local ones, the authors obtain (GCR) in general smooth domains through localization, and they provide auxiliary tools such as sup-convolution regularization and convexity-preserving subsolution constructions. The results enable Perron-based existence and give insight into boundary behavior and regularization, with open questions on regularity and large-time behavior.

Abstract

In this article, we consider fully nonlinear, possibly degenerate, parabolic equations associated with Ventcell boundary conditions in bounded or unbounded, smooth domains. We first analyze the exact form of such boundary conditions in general domains in order that the notion of viscosity solutions makes sense. Then we prove general comparison results, both for first- and second-order equations, under rather natural assumptions on the nonlinearities: in the second-order case, the only restrictive assumption is that the equation has to be strictly elliptic in the normal direction, in a neighborhood of the boundary; in the first-order one, quasiconvexity assumptions have to be imposed both on the equation and the boundary condition, the equation being coercive in the normal direction. Our method is inspired by the ``twin blow-up method'' of Forcadel-Imbert-Monneau, that we adapt to a scaling consistent with the Ventcell boundary condition.
Paper Structure (17 sections, 16 theorems, 197 equations)

This paper contains 17 sections, 16 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. The Ventcell Boundary Condition in General Domains
  3. Consistency with the Maximum Principle
  4. Reduction to a flat comparison result
  5. Preliminaries
  6. Hypotheses
  7. Global Comparison Results from Local Comparison Results
  8. Local Properties of the Ventcell Boundary Condition
  9. About the initial condition
  10. Reduced parabolic sub- and superjets
  11. Statement of the Main Comparison Results
  12. Proof of (LCR) in the Half-Space Case in the First-Order Case
  13. Proof of (LCR) in the Half-Space Case in the Second-Order Case
  14. Further Results and Open Questions
  15. Computation of sup-convolutions
  16. ...and 2 more sections

Key Result

Proposition 2.1

Let $x\in {\partial \Omega}$ be a local maximum point on ${\overline \Omega}$ of $y\mapsto (u-\phi)(y)$. Then the following first and second-order inequalities hold:

Theorems & Definitions (19)

  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Remark 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Proposition 3.6
  • ...and 9 more