Generalized junction conditions for degenerate parabolic equations

Cyril Imbert (1) , Vinh Duc Nguyen (2)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

We are interested in the study of parabolic equations on a multi-dimensional junction (Imbert, Monneau (2014)), i.e. the union of a finite number of copies of a half-hyperplane of R d+1 whose boundaries are identified. The common boundary is referred to as the junction hyperplane. The parabolic equations on the half-hyperplanes are in non-divergence form, fully non-linear and possibly degenerate, and they do degenerate along the junction hyperplane, i.e. along the junction hyperplane the nonlinearities do not depend on second order derivatives. The parabolic equations are supplemented with a generalized junction condition (or boundary condition of Neumann type), which is compatible with the maximum principle. Our main result states that, in the case where the non-linearities at the junction have convex sublevel sets with respect to the gradient variable, then these general junction conditions can be classified: they are equivalent to junction conditions of control type. This classification extends the one obtained by Imbert and Monneau for Hamilton-Jacobi equations on networks and multi-dimensional junctions. We give two applications of this classification result. On the one hand, we give the first complete answer to an open question about these equations: we prove in the two-domain case that the vanishing viscosity limit associated with quasi-convex Hamilton-Jacobi equations coincides with the maximal Ishii solution identified by Barles, Briani and Chasseigne (2012). On the other hand, we give a short and simple PDE proof of a large deviation results of Boué, Dupuis and Ellis (2000).

Domaines: Equations aux dérivées partielles [math.AP]

Cyril Imbert, Vinh Duc Nguyen. Generalized junction conditions for degenerate parabolic equations. 2016. ⟨hal-01252891v2⟩