The twin blow-up method for Hamilton-Jacobi equations in higher dimension

TL;DR

This paper extends the twin blow-up method to higher-dimensional Hamilton-Jacobi equations, enabling a new comparison principle that handles complex boundary conditions and variable coupling.

Contribution

It introduces a novel dual blow-up technique for Hamilton-Jacobi equations, improving the analysis of boundary behavior in higher dimensions.

Findings

Developed a new comparison principle for Hamilton-Jacobi equations in any dimension.

Extended the twin blow-up method to handle boundary coupling of variables.

Established a one-sided Lipschitz estimate for blow-up limits.

Abstract

In this paper, we show how to extend the twin blow-up method recently developped by the authors (Comptes Rendus. Math., 2024), in order to obtain a new comparison principle for an evolution coercive Hamilton-Jacobi equation posed in a domain of an Euclidian space of any dimension and supplemented with a boundary condition. The method allows dealing with the case where tangential variables and the variable corresponding to the normal gradient of the solution are strongly coupled at the boundary. We elaborate on a method introduced by P.-L. Lions and P. Souganidis (Atti Accad. Naz. Lincei, 2017). Their argument relies on a single blow-up procedure after rescaling the semi-solutions to be compared while two simultaneous blow-ups are performed in this work, one for each variable of the classical doubling variable technique. A one-sided Lipschitz estimate satisfied by a combination of the two blow-up limits plays a key role.