Effective nonlinear Neumann boundary conditions for 1D nonconvex Hamilton–Jacobi equations

Abstract

We study Hamilton-Jacobi equations in [0, +∞) of evolution type with nonlinear boundary conditions of Neumann type in the case where the Hamiltonian is non necessarily convex with respect to the gradient variable. In this paper, we give two main results. First, we prove a classification of boundary condition result for a nonconvex, coercive Hamiltonian, in the spirit of the flux-limited formulation for quasi-convex Hamilton-Jacobi equations on networks recently introduced by Imbert and Monneau. Second, we give a comparison principle for a nonconvex and noncoercive Hamiltonian where the boundary condition can have flat parts. Mathematics Subject Classification: 49L25, 35B51, 35F30, 35F21This problem is an extension to the state constraint problem of Soner [24] and Ishii and Koike [20], where the authors study the case of a convex Hamiltonian. For H quasiconvex, in [17], the authors prove that (2) is equivalent towhere H − is the decreasing part of the Hamiltonian, see also [13] for the multidimensional case. If we define for H nonconvex,

Effective junction conditions for degenerate parabolic equations

“…On the one hand, the proof was quite difficult, relying on the vertex test function introduced in [18,17], for which C 2 regularity was to be proved in the multidimensional setting. On the other hand, new and simpler techniques now emerge to attack this problem, see for instance [7,16,23,24]. In particular, it is explained in [7] that the equations considered in the present work can be handled in the two-domain case.…”

“…We also assume that the Hamiltonians have convex sublevel sets, see (F4). This condition can probably be relaxed but until very recent contributions [16,24,23] (none of these contributions were not available when the first version of this work appeared), the non-convex case was out of reach. As far as (F3) is concerned, it ensures that the Hamiltonians are coercive, a property which is used repeatedly and is at the core of most proofs.…”

“…We previously mentioned that, since the first version of this paper were posted, Guerand and Monneau studied independently effective non-linear boundary conditions in the nonconvex case. On the one hand Guerand [16] studied the case N = 1 in the 1D setting, which amounts to studying first order non-convex Hamilton-Jacobi equations with nonlinear boundary conditions of Neumann type. On the other hand Monneau [24] mentioned to us that he studies effective junction conditions for non-convex Hamilton-Jacobi equations posed on multi-dimensional junctions.…”

A comparison principle for Hamilton-Jacobi equation with moving in time boundary

“…Let us mention also the work [28] where the authors consider a Kirchoff-type Neumann condition at the junction and proved that its solution satisfy a comparison principle and then they proved that the fluxlimited solutions reduce to Kirchoff-type viscosity solutions. Finally, concerning comparison principle for Hamilton-Jacobi equations with boundary conditions of Neumann type, let us cite [27,1,5,19,2,13,23]. Combaining the comparison principle for (1) with Perron method, we obtain the following main result Theorem 1.1.…”

Homogenization results for a deterministic multi-domains periodic control problem

“…Therefore it seems that such particular definitions have to be used in each case since, again, the Kirchhoff condition does not seem natural in the control framework. Remark 7.2.2 Definition 7.2.1 provides the notion of "flux-limited viscosity solutions" for a problem with a co-dimension 1 discontinuity but it can be used in different frameworks, in particular in problems with boundary conditions: we refer to Guerand [61] for results on state constraints problems and [60] in the case of Neumann conditions where "effective boundary conditions and new comparison results are given, both works being in the case of quasi-convex Hamiltonians.…”