A Differential Game with Two Players and One Target: The Continuous Case

We study a two-players differential game in which one player wants the state of the system to reach an open target while the other player wants the state of the system to avoid this target. The aim of this paper is to show that, if the first player plays "Caratheodory strategies" and the second player plays controls, then the game is not well-defined, i.e., either the "alternative" or the "causality" is not satisfied for that game

A Differential Game With One Target and Two Players

We study a two players-differential game in which one player wants the state of the system to reach an open target and the other player wants the state of the system to avoid the target. We characterize the victory domains of each player as the largest set satisfying some geometric conditions and we show a "barrier phenomenon" on the boundary of the victory domains

A Non-Local Mean Curvature Flow and its semi-implicit time-discrete approximation

We address in this paper the study of a geometric evolution, corresponding to a curvature which is non-local and singular at the origin. The curvature represents the first variation of the energy recently proposed as a variant of the standard perimeter penalization for the denoising of nonsmooth curves. To deal with such degeneracies, we first give an abstract existence and uniqueness result for viscosity solutions of non-local degenerate Hamiltonians, satisfying suitable continuity assumption with respect to Kuratowsky convergence of the level sets. This abstract setting applies to an approximated flow. Then, by the method of minimizing movements, we also build an "exact" curvature flow, and we illustrate some examples, comparing the results with the standard mean curvature flow

Global Existence Results and Uniqueness for Dislocation Equations

We are interested in nonlocal Eikonal Equations arising in the study of the dynamics of dislocations lines in crystals. For these nonlocal but also non monotone equations, only the existence and uniqueness of Lipschitz and local-in-time solutions were available in some particular cases. In this paper, we propose a definition of weak solutions for which we are able to prove the existence for all time. Then we discuss the uniqueness of such solutions in several situations, both in the monotone and non monotone case

Representation of equilibrium solutions to the table problem for growing sandpiles

In the dynamical theory of granular matter the so-called table problem consists in studying the evolution of a heap of matter poured continuously onto a bounded domain Omega subset of R-2. The mathematical description of the table problem, at an equilibrium configuration, can be reduced to a boundary value problem for a system of partial differential equations. The analysis of such a system, also connected with other mathematical models such as the Monge-Kantorovich problem, is the object of this paper. Our main result is an integral representation formula for the solution, in terms of the boundary curvature and of the normal distance to the cut locus of Omega

Regularity properties of attainable sets under state constraints

The Maximum principle in control theory provides necessary optimality conditions for a given trajectory in terms of the co-state, which is the solution of a suitable adjoint system. For constrained problems the adjoint system contains a measure supported at the boundary of the constraint set. In this paper we give a representation formula for such a measure for smooth constraint sets and nice Hamiltonians. As an application, we obtain a perimeter estimate for constrained attainable sets

A boundary value problem for a PDE model in mass transfer theory: Representation of solutions and applications

The system of partial differential equations -div (vDu) = f in Q vertical bar Du vertical bar - 1 = 0 in {v > 0} arises in the analysis of mathematical models for sandpile growth and in the context of the Monge-Kantorovich optimal mass transport theory. A representation formula for the solutions of a related boundary value problem is here obtained, extending the previous two-dimensional result of the first two authors to arbitrary space dimension. An application to the minimization of integral functionals of the form integral(Omega) [h(vertical bar Du vertical bar)-f(x)u]dx, with f >= 0, and h >= 0 possibly non-convex, is also included

Remarks on Nash equilibria in mean field game models with a major player

For a mean field game model with a major and infinite minor players, we characterize a notion of Nash equilibrium via a system of so-called master equations, namely a system of nonlinear transport equations in the space of measures. Then, for games with a finite number N of minor players and a major player, we prove that the solution of the corresponding Nash system converges to the solution of the system of master equations as N tends to infinity

Some flows in shape optimization

Geometric flows related to shape optimization problems of Bernoulli type are investigated. The evolution law is the sum of a curvature term and a nonlocal term of Hele-Shaw type. We introduce generalized set solutions, the definition of which is widely inspired by viscosity solutions. The main result is an inclusion preservation principle for generalized solutions. As a consequence, we obtain existence, uniqueness and stability of solutions. Asymptotic behavior for the flow is discussed: we prove that the solutions converge to a generalized Bernoulli exterior free boundary problem