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.

In this article, we are interested in semilinear, possibly degenerate elliptic equations posed on a general network, with nonlinear Kirchhoff-type conditions for its interior vertices and Dirichlet boundary conditions for the boundary ones. The novelty here is the generality of the equations posed on each edge that is incident to a particular vertex, ranging from first-order equations to uniformly elliptic ones. Our main result is a strong comparison principle, i.e., a comparison result between discontinuous viscosity sub and supersolutions of such problems, from which we conclude the existence and uniqueness of a continuous viscosity by Perron's method. Further extensions are also discussed.

In this article, we consider nonlocal Hamilton-Jacobi Equations on networks with Kirchhoff type conditions for the interior vertices and Dirichlet boundary conditions for the boundary ones: our aim is to provide general existence and comparison results in the case when the integro-differential operators are of order strictly less than 1. The main originality of these results is to allow these nonlocal terms to have contributions on several different edges of the network. The existence of Lipschitz continuous solutions is proved in two ways: either by using the vanishing viscosity method or by the usual Perron's method. The comparison proof relies on arguments introduced by Lions and Souganidis. We also introduce a notion of flux-limited solution, nonlocal analog to the one introduced by Imbert and Monneau, and prove that the solutions of the Kirchhoff problem are flux-limited solutions for a suitable flux-limiter. After treating in details the case when we only have one interior vertex, we extend our approach to treat general networks.

This version is the last version of our book project on Hamilton-Jacobi Equations and Control Problems with discontinuities. Compared to the third version (online in december 2022), we have improved Part V (Stratified solutions for state-constraints problems) and Part VI on the applications but also the stability results for stratified solutions; we have rewritten a large part of the introduction and added guidelines for the reader. As in the previous versions, we have incorporated new results and examples, changed some points-of-view, detailed some proofs and corrected several mistakes. Version 3 had 550 pages, this one 630.As the third version, it is composed of six parts: Part I is still a toolbox with key results which are used in all the other parts. The study of the simplest case, i.e. the case of a co-dimension 1 discontinuity, is now split in two parts: in Part II, we only consider control problems and the associated Bellman Equations are treated by using only the classical notion of viscosity solutions. In this part, the methods are a combinations of control and pdes techniques. On the contrary, Part III describes purely pdes approaches which are inspired by the literature on Hamilton Jacobi Equations on networks and which can handle the case of non-convex Hamiltonians. In this part, we present two notions of solutions, namely flux-limited and junction viscosity solutions, and we study in detail their properties by providing comparison and stability results. We also show that they are ``almost'' equivalent when both make sense, i.e. for quasi-convex Hamiltonians. Part IV concerns stratified problems in $\R^N$, i.e. problems with discontinuities of any co-dimensions: the main change compared to the previous version is the introduction of a notion of ``weak'' stratified (sub)solution. In Part V, we address the case of stratified problems in bounded or unbounded domains with state-constraints, allowing very surprising applications as well as singular boundary conditions. Finally, in Part VI we describe some applications to KPP (Kolmogorov-Petrovsky-Piskunov) type problems and we discuss possible extensions to problems with jumps and to ``stratified networks''.Even if we consider this version as being the final one, all comments are welcome!