  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.
