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Global subanalytic solutions of Hamilton–Jacobi type equations. (English) Zbl 1094.35020

In this work, the Dirichlet and Cauchy-Dirichlet problems for Hamilton-Jacobi equations are investigated. The Hamiltonian function is associated to an analytic Lagrangian function and the existence, and in some cases, the uniqueness of a subanalytic viscosity solution is established. These statements are then extended to cases where the Hamiltonian function is stemming from sub-Riemannian geometry, and more generally from an optimal control problem, and is showed how the subanalyticity status of solutions is related to the existence of singular minimizing trajectories of the underlying control problem. The results of this paper give conditions under which the vioscosity solution of some Hamilton-Jacobi equations is subanalytic. These results imply that the cut-locus, which coincides with the analytic singular set of the viscosity solution, is a subanalytic stratified manifold of codimension greater than or equal to one. Note that this singular set is also the set where characteristic curves intersect. This property is very useful in numerical analysis and has already been used. The interest is to get a general framework in which the set where characteristic curves intersect is, in a sense, “small”. Usual methods to derive such a fact rely on a careful analysis of the characteristic curves, that may be very involved. The results of this paper provide systematic sufficient conditions under which this singular set shares these nice properties resulting from subanalyticity.

MSC:

35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35F20 Nonlinear first-order PDEs
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
32B20 Semi-analytic sets, subanalytic sets, and generalizations
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