Olivier Alvarez

CONVEX VISCOSITY SOLUTIONS AND STATE CONSTRAINTS

Abstract We establish the convexity of a viscosity solution of some general second order fully nonlinear elliptic equation with state constraints boundary conditions. Our method combines a comparison principle with the observation that, under suitable assumptions, the convex envelope of the solution is a supersolution. This property relies on the characterization of the viscosity subject of the convex envelope of a lower semicontinuous coercive function. The equation solved by the conjugate of a convex solution as well as partial convexity are topics we also discuss.

Viscosity solutions of nonlinear integro-differential equations

Abstract We investigate the questions of the existence and uniqueness of viscosity solutions to the Cauchy problem for integro-differential PDEs with nonlinear integral term. The existence of a solution is established by considering semicontinuous subsolutions and supersolutions and applying Perron’s method. Uniqueness is proved for both bounded and unbounded solutions. These results are then applied to a problem arising in Finance, namely the stochastic differential utility model under mixed Poisson-Brownian information.

Viscosity Solutions Methods for Singular Perturbations in Deterministic and Stochastic Control

Viscosity solutions methods are used to pass to the limit in some penalization problems for first order and second order, degenerate parabolic, Hamilton--Jacobi--Bellman equations. This characterizes the limit of the value functions of singularly perturbed optimal control problems for deterministic systems and for controlled degenerate diffusions. The results apply to cases where the usual order reduction method does not give the correct limit, and to systems with fast state variables depending nonlinearly on the control. Some connections with ergodic control and periodic homogenization are discussed.

Existence and uniqueness for dislocation dynamics with nonnegative velocity

We study the problem of large time existence of solutions for a mathematical model describing dislocation dynamics in crystals. The mathematical model is a geometric and nonlocal eikonal equation which does not preserve the inclusion. Under the assumption that the dislocation line is expanding, we prove existence and uniqueness of the solution in the framework of discontinuous viscosity solutions. We also show that this solution satisfies some variational properties, which allows us to prove that the energy associated to the dislocation dynamics is nonincreasing.

Ergodic Problems in Differential Games

We present and study a notion of ergodicity for deterministic zero-sum differential games that extends the one in classical ergodic control theory to systems with two conflicting controllers.We show its connections with the existence of a constant and uniform long-time limit of the value function of finite horizon games, and characterize this property in terms of Hamilton-Jacobi-Isaacs equations.We also give several sufficient conditions for ergodicity and describe some extensions of the theory to stochastic differential games.

A convergent scheme for a non local Hamilton Jacobi equation modelling dislocation dynamics

We study dislocation dynamics with a level set point of view. The model we present here looks at the zero level set of the solution of a non local Hamilton Jacobi equation, as a dislocation in a plane of a crystal. The front has a normal speed, depending on the solution itself. We prove existence and uniqueness for short time in the set of continuous viscosity solutions. We also present a first order finite difference scheme for the corresponding level set formulation of the model. The scheme is based on monotone numerical Hamiltonian, proposed by Osher and Sethian. The non local character of the problem makes it not monotone. We obtain an explicit convergence rate of the approximate solution to the viscosity solution. We finally provide numerical simulations.

HAMILTON-JACOBI EQUATIONS WITH PARTIAL GRADIENT AND APPLICATION TO HOMOGENIZATION

The paper proves that the Dirichlet problem for the first-order Hamilton-Jacobi equation in an open subset of R n where D x′ u is the partial gradient of the scalar function u with respect to the first n′ variables (n′ ≤ n), has a viscosity solution which is unique a.e. When applied to the periodic homogenization of Hamilton-Jacobi equations in a perforated set, the result yields the a.e. convergence of the solutions of the problem at scale ε as ε → 0 to the solution of the homogenized Hamilton-Jacobi equation. *Supported in part by the TMR Network “Viscosity Solutions and Applications.” †Supported in part by Grant-in-Aid for Scientific Research, No. 09440067, Ministry of Education, Science, Sports and Culture.

Ergodic Control in L

The infinite horizon discounted L∞ problem is studied as the discount factor goes to zero. It is related to the limit of the finite horizon problem as the horizon becomes infinite. The deterministic problem with and without a running cost is considered.

Some ergodic problems for differential games

We present a notion of ergodicity for deterministic zero-sum differential games that extends the one in classical ergodic control theory to systems with two conflicting controllers. We describe its connections with the existence of a constant and uniform long-time limit of the value function of finite horizon games, and characterize this property in terms of Hamilton-Jacobi-Isaacs equations. We also give several sufficient conditions for ergodicity and describe some extensions of the theory to stochastic differential games.