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Pierre Cardaliaguet

Publications and source records attributed to Pierre Cardaliaguet.

67 records · Page 4Linked to original sources
arXiv2010

A differential game with a blind player

We consider a zero sum differential game with lack of observation on one side. The initial state of the system is drawn at random according to some probability $μ_0$ on $\R^N$. Player-I is informed of the initial position of state while player-II knows only $μ_0$. Moreover Player-I observes Player-II's moves while Player-II is blind and has no further information. We prove that in this game with a terminal payoff the value exists and is characterized as the unique viscosity solution of some Hamilton-Jacobi equation on a space of probability measures.

math.OC↗
arXiv2010

Homogenization and enhancement for the G-equation

We consider the so-called G-equation, a level set Hamilton-Jacobi equation, used as a sharp interface model for flame propagation, perturbed by an oscillatory advection in a spatio-temporal periodic environment. Assuming that the advection has suitably small spatial divergence, we prove that, as the size of the oscillations diminishes, the solutions homogenize (average out) and converge to the solution of an effective anisotropic first-order (spatio-temporal homogeneous) level set equation. Moreover we obtain a rate of convergence and show that, under certain conditions, the averaging enhances the velocity of the underlying front. We also prove that, at scale one, the level sets of the solutions of the oscillatory problem converge, at long times, to the Wulff shape associated with the effective Hamiltonian. Finally we also consider advection depending on position at the integral scale.

math.AP↗
arXiv2010

Viscosity solutions for a polymer crystal growth model

We prove existence of a solution for a polymer crystal growth model describing the movement of a front $(Γ(t))$ evolving with a nonlocal velocity. In this model the nonlocal velocity is linked to the solution of a heat equation with source $δ_Γ$. The proof relies on new regularity results for the eikonal equation, in which the velocity is positive but merely measurable in time and with Hölder bounds in space. From this result, we deduce \textit{a priori} regularity for the front. On the other hand, under this regularity assumption, we prove bounds and regularity estimates for the solution of the heat equation.

math.AP↗
arXiv2010

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.

math.AP↗
arXiv2010

Hölder regularity for viscosity solutions of fully nonlinear, local or nonlocal, Hamilton-Jacobi equations with super-quadratic growth in the gradient

Viscosity solutions of fully nonlinear, local or non local, Hamilton-Jacobi equations with a super-quadratic growth in the gradient variable are proved to be Hölder continuous, with a modulus depending only on the growth of the Hamiltonian. The proof involves some representation formula for nonlocal Hamilton-Jacobi equations in terms of controlled jump processes and a weak reverse inequality.

math.OC↗
arXiv2009

Hölder estimates in space-time for viscosity solutions of Hamilton-Jacobi equations

It is well-known that solutions to the basic problem in the calculus of variations may fail to be Lipschitz continuous when the Lagrangian depends on t. Similarly, for viscosity solutions to time-dependent Hamilton-Jacobi equations one cannot expect Lipschitz bounds to hold uniformly with respect to the regularity of coefficients. This phenomenon raises the question whether such solutions satisfy uniform estimates in some weaker norm. We will show that this is the case for a suitable Hölder norm, obtaining uniform estimates in (x,t) for solutions to first and second order Hamilton-Jacobi equations. Our results apply to degenerate parabolic equations and require superlinear growth at infinity, in the gradient variables, of the Hamiltonian. Proofs are based on comparison arguments and representation formulas for viscosity solutions, as well as weak reverse Hölder inequalities.

math.OC↗
arXiv2009

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.

math.AP↗
arXiv2008

On a continuous time game with incomplete information

For zero-sum two-player continuous-time games with integral payoff and incomplete information on one side, one shows that the optimal strategy of the informed player can be computed through an auxiliary optimization problem over some martingale measures. One also characterizes the optimal martingale measures and compute it explicitely in several examples.

math.PR↗
arXiv2008

Existence of weak solutions for general nonlocal and nonlinear second-order parabolic equations

In this article, we provide existence results for a general class of nonlocal and nonlinear second-order parabolic equations. The main motivation comes from front propagation theory in the cases when the normal velocity depends on the moving front in a nonlocal way. Among applications, we present level-set equations appearing in dislocations' theory and in the study of Fitzhugh-Nagumo systems.

math.AP↗
arXiv2008

Uniqueness Results for Nonlocal Hamilton-Jacobi Equations

We are interested in nonlocal Eikonal Equations describing the evolution of interfaces moving with a nonlocal, non monotone velocity. For these equations, only the existence of global-in-time weak solutions is available in some particular cases. In this paper, we propose a new approach for proving uniqueness of the solution when the front is expanding. This approach simplifies and extends existing results for dislocation dynamics. It also provides the first uniqueness result for a Fitzhugh-Nagumo system. The key ingredients are some new perimeter estimates for the evolving fronts as well as some uniform interior cone property for these fronts.

math.AP↗
arXiv2007

Stochastic differential games with asymmetric information

We investigate a two-player zero-sum stochastic differential game in which the players have an asymmetric information on the random payoff. We prove that the game has a value and characterize this value in terms of dual solutions of some second order Hamilton-Jacobi equation.

math.OC↗
arXiv2006

On the energy of a flow arising in shape optimization

In Cardaliaguet-Ley (2006) we have defined a viscosity solution for the gradient flow of the exterior Bernoulli free boundary problem. We prove here that the associated energy is non decreasing along the flow. This justifies the "gradient flow" approach for such kind of problem. The proof relies on the construction of a discrete gradient flow in the flavour of Almgren-Taylor-Wang (1993) and on proving it converges to the viscosity solution.

math.AP↗