Variational methods for Hamilton-Jacobi equations and applications.

Summary The objective of this thesis is to propose variational methods for the mathematical and numerical analysis of a class of HJ equations. The metric character of these equations allows to characterize the set of sub-solutions, namely, they are 1-Lipschitz with respect to the Finslerian distance associated to the Hamiltonian. Equivalently, this is equivalent to saying that the gradient of these functions belongs to a certain Finslerian ball. The solution we are looking for is the maximal subsolution, which can be described by a Hopf-Lax type formula, which solves a maximization problem with constraint on the gradient. We derive an associated dual problem involving the total Finslerian variation of vector measures with divergent constraint. We exploit the saddle-point structure to propose a numerical solution with the augmented Lagrangian method. This characterization of the HJ equation also shows the link with optimal transport problems to/from the edge. This link with optimal mass transport leads us to generalize the Evans-Gangbo approach. Indeed, we show that the maximal subsolution of the HJ equation is obtained by stretching p→∞ in a class of Finsler-type p-Laplacians with obstacles on the edge. This also allows us to construct the optimal flow for the associated Beckmann problem. Among the applications we look at is the Shape from Shading problem, which involves reconstructing the surface of a 3D object from a grayscale image of that object.