Variational Methods for Hamilton-Jacobi Equations and Applications

Book excerpt: In this thesis we propose some variational methods for the mathematical and numerical analysis of a class of HJ equations. Thanks to the metric character of these equations, the set of subsolution corresponds to the set of 1-Lipschitz functions with respect to the Finsler metric associated to the Hamiltonian. Equivalently, it corresponds to the set of functions whose gradient belongs to a Finsler ball. The solution we are looking for is the maximal one, which can be described via a Hopf-Lax formula, solves a maximization problem under gradient constraint. We derive the associated dual problem which involves the Finsler total variation of vector measures under a divergence constraint. We take advantage of this saddle-point structure to use the augmented Lagrangian method for the numerical approximation of HJ equation. This characterization of the HJ equation allows making the link with some optimal transport problems. This link with optimal transport leads us to generalize the Evans-Gangbo approach. In fact, we show that the maximal viscosity subsolution of the HJ equation can be recovered by taking p→ ∞ in a class of Finslerp-Laplace problems with boundary obstacles. In addition, this allows us to construct the optimal flow for the associated Beckmann problem. As an application, we use our variational approach for the Shape from Shading problem.