arXiv subjects

Emmanuel Chasseigne

Publications and source records attributed to Emmanuel Chasseigne.

At least 19 recordsLinked to original sources

arXiv2024

Unbounded Hamilton-Jacobi-Bellman Equations with one co-dimensional discontinuities

The aim of this work is to deal with a discontinuous Hamilton-Jacobi equation in the whole euclidian N-dimensional space, associated to a possibly unbounded optimal control problem. Here, the discontinuities are located on a hyperplane and the typical questions we address concern the existence and uniqueness of solutions, and of course the definition itself of solution. We consider viscosity solutions in the sense of Ishii. The convex Hamiltonians are associated to a control problem with specific cost and dynamics given on each side of the hyperplane. We assume that those are Lipschitz continuous but the main difficulty we deal with is that they are potentially unbounded, as well as the control spaces. Using Bellman's approach we construct two value functions which turn out to be the minimal and maximal solutions in the sense of Ishii. Moreover, we also build a whole family of value functions, which are still solutions in the sense of Ishii and connect continuously the minimal solution to the maximal one.

math.OC↗

arXiv2024

Some Comparison Results for First-Order Hamilton-Jacobi Equations and Second-Order Fully Nonlinear Parabolic Equations with Ventcell Boundary Conditions

In this article, we consider fully nonlinear, possibly degenerate, parabolic equations associated with Ventcell boundary conditions in bounded or unbounded, smooth domains. We first analyze the exact form of such boundary conditions in general domains in order that the notion of viscosity solutions makes sense. Then we prove general comparison results, both for first- and second-order equations, under rather natural assumptions on the nonlinearities: $(i)$ in the second-order case, the only restrictive assumption is that the equation has to be strictly elliptic in the normal direction, in a neighborhood of the boundary; $(ii)$ in the first-order one, quasiconvexity assumptions have to be imposed both on the equation and the boundary condition, the equation being coercive in the normal direction. Our method is inspired by the ``twin blow-up method'' of Forcadel-Imbert-Monneau, that we adapt to a scaling consistent with the Ventcell boundary condition.

math.AP↗

arXiv2023

An Illustrated Guide of the Modern Approaches of Hamilton-Jacobi Equations and Control Problems with Discontinuities

This version is the last version of our book project on Hamilton-Jacobi Equations and Control Problems with discontinuities. Compared to the third version (online in december 2022), we have improved Part V (Stratified solutions for state-constraints problems) and Part VI on the applications but also the stability results for stratified solutions; we have rewritten a large part of the introduction and added guidelines for the reader. As in the previous versions, we have incorporated new results and examples, changed some points-of-view, detailed some proofs and corrected several mistakes. Version 3 had 550 pages, this one 630.As the third version, it is composed of six parts: Part I is still a toolbox with key results which are used in all the other parts. The study of the simplest case, i.e. the case of a co-dimension 1 discontinuity, is now split in two parts: in Part II, we only consider control problems and the associated Bellman Equations are treated by using only the classical notion of viscosity solutions. In this part, the methods are a combinations of control and pdes techniques. On the contrary, Part III describes purely pdes approaches which are inspired by the literature on Hamilton Jacobi Equations on networks and which can handle the case of non-convex Hamiltonians. In this part, we present two notions of solutions, namely flux-limited and junction viscosity solutions, and we study in detail their properties by providing comparison and stability results. We also show that they are ``almost'' equivalent when both make sense, i.e. for quasi-convex Hamiltonians. Part IV concerns stratified problems in $\R^N$, i.e. problems with discontinuities of any co-dimensions: the main change compared to the previous version is the introduction of a notion of ``weak'' stratified (sub)solution. In Part V, we address the case of stratified problems in bounded or unbounded domains with state-constraints, allowing very surprising applications as well as singular boundary conditions. Finally, in Part VI we describe some applications to KPP (Kolmogorov-Petrovsky-Piskunov) type problems and we discuss possible extensions to problems with jumps and to ``stratified networks''.Even if we consider this version as being the final one, all comments are welcome!

math.AP↗

arXiv2019

Sharp estimates of the generalized principal eigenvalue for superlinear viscous Hamilton-Jacobi equations with inward drift

This paper is concerned with the ergodic problem for viscous Hamilton-Jacobi equations having superlinear Hamiltonian, inward-pointing drift, and positive potential which vanishes at infinity. Assuming some radial symmetry of the drift and the potential outside a ball, we establish sharp estimates of the generalized principal eigenvalue with respect to a perturbation of the potential.

math.AP↗

arXiv2018

On unbounded solutions of ergodic problems for non-local Hamilton-Jacobi equations

We study an ergodic problem associated to a non-local Hamilton-Jacobi equation defined on the whole space $λ-\mathcal{L}[u](x)+|Du(x)|^m=f(x)$ and determine whether (unbounded) solutions exist or not. We prove that there is a threshold growth of the function $f$, that separates existence and non-existence of solutions, a {phenomenon} that does not appear in the local version of the problem. Moreover, we show that there exists a critical ergodic constant, $λ_*$, such that the ergodic problem has solutions for $λ\leq λ_*$ and such that the only solution bounded from below, which is unique up to an additive constant, is the one associated to $λ_*$.

math.AP↗

arXiv2017

A Priori Lipschitz Estimates for Solutions of Local and Nonlocal Hamilton-Jacobi Equations with Ornstein-Uhlenbeck Operator

We establish a priori Lipschitz estimates for unbounded solutions of second-order Hamilton-Jacobi equations in R^N in presence of an Ornstein-Uhlenbeck drift. We generalize the results obtained by Fujita, Ishii \& Loreti (2006) in several directions. The first one is to consider more general operators. We first replace the Laplacian by a general diffusion matrix and then consider a nonlocal integro-differential operator of fractional Laplacian type. The second kind of extension is to deal with more general Hamiltonians which are merely sublinear. These results are obtained for both degenerate and nondegenerate equations.

math.AP↗

arXiv2016

On nonlocal quasilinear equations and their local limits

We introduce a new class of quasilinear nonlocal operators and study equations involving these operators. The operators are degenerate elliptic and may have arbitrary growth in the gradient. Included are new nonlocal versions of p-Laplace, $\infty$-Laplace, mean curvature of graph, and even strongly degenerate operators, in addition to some nonlocal quasilinear operators appearing in the existing literature. Our main results are comparison, uniqueness, and existence results for viscosity solutions of linear and fully nonlinear equations involving these operators. Because of the structure of our operators, especially the existence proof is highly non-trivial and non-standard. We also identify the conditions under which the nonlocal operators converge to local quasilinear operators, and show that the solutions of the corresponding nonlocal equations converge to the solutions of the local limit equations. Finally, we give a (formal) stochastic representation formula for the solutions and provide many examples.

math.AP↗

arXiv2016

Flux-limited and classical viscosity solutions for regional control problems

The aim of this paper is to compare two different approaches for regional control problems: the first one is the classical approach, using a standard notion of viscosity solutions, which is developed in a series of works by the three first authors. The second one is more recent and relies on ideas introduced by Monneau and the fourth author for problems set on networks in another series of works, in particular the notion of flux-limited solutions. After describing and even revisiting these two very different points of view in the simplest possible framework, we show how the results of the classical approach can be interpreted in terms of flux-limited solutions. In particular, we give much simpler proofs of three results: the comparison principle in the class of bounded flux-limited solutions of stationary multidimensional Hamilton-Jacobi equations and the identification of the maximal and minimal Ishii's solutions with flux-limited solutions which were already proved by Monneau and the fourth author, and the identification of the corresponding vanishing viscosity limit, already obtained by Vinh Duc Nguyen and the fourth author.

math.AP↗

arXiv2016

Qualitative properties of generalized principal eigenvalues for superquadratic viscous Hamilton-Jacobi equations

This paper is concerned with the ergodic problem for superquadratic viscous Hamilton-Jacobi equations with exponent m \textgreater{} 2. We prove that the generalized principal eigenvalue of the equation converges to a constant as m $\rightarrow$ $\infty$, and that the limit coincides with the generalized principal eigenvalue of an ergodic problem with gradient constraint. We also investigate some qualitative properties of the generalized principal eigenvalue with respect to a perturbation of the potential function. It turns out that different situations take place according to m = 2, 2 \textless{} m \textless{} $\infty$, and the limiting case m = $\infty$.

math.AP↗

arXiv2015

On the regularizing effect for unbounded solutions of first-order Hamilton-Jacobi equations

We give a simplified proof of regularizing effects for first-order Hamilton-Jacobi Equations of the form $u\_t+H(x,t,Du)=0$ in $\R^N\times(0,+\infty)$ in the case where the idea is to first estimate $u\_t$. As a consequence, we have a Lipschitz regularity in space and time for coercive Hamiltonians and, for hypo-elliptic Hamiltonians, we also have an H\''older regularizing effect in space following a result of L. C. Evans and M. R. James.

math.AP↗

arXiv2015

(Almost) Everything You Always Wanted to Know About Deterministic Control Problems in Stratified Domains

We revisit the pioneering work of Bressan \& Hong on deterministic control problems in stratified domains, i.e. control problems for which the dynamic and the cost may have discontinuities on submanifolds of R N . By using slightly different methods, involving more partial differential equations arguments, we (i) slightly improve the assumptions on the dynamic and the cost; (ii) obtain a comparison result for general semi-continuous sub and supersolutions (without any continuity assumptions on the value function nor on the sub/supersolutions); (iii) provide a general framework in which a stability result holds.

math.AP↗

arXiv2014

Homogenization Results for a Deterministic Multi-domains Periodic Control Problem

We consider homogenization problems in the framework of deterministic optimal control when the dynamics and running costs are completely different in two (or more) complementary domains of the space $\R^N$. For such optimal control problems, the three first authors have shown that several value functions can be defined, depending, in particular, of the choice is to use only "regular strategies" or to use also "singular strategies". We study the homogenization problem in these two different cases. It is worth pointing out that, if the second one can be handled by usual partial differential equations method " á la Lions-Papanicolaou-Varadhan" with suitable adaptations, the first case has to be treated by control methods (dynamic programming).

math.AP↗

arXiv2014

A Bellman approach for regional optimal control problems in $\R^N$

This article is a continuation of a previous work where we studied infinite horizon control problems for which the dynamic, running cost and control space may be different in two half-spaces of some euclidian space $\R^N$. In this article we extend our results in several directions: $(i)$ to more general domains; $(ii)$ by considering finite horizon control problems; $(iii)$ by weaken the controlability assumptions. We use a Bellman approach and our main results are to identify the right Hamilton-Jacobi-Bellman Equation (and in particular the right conditions to be put on the interfaces separating the regions where the dynamic and running cost are different) and to provide the maximal and minimal solutions, as well as conditions for uniqueness. We also provide stability results for such equations.

math.AP↗

arXiv2013

Fractional decay bounds for nonlocal zero order heat equations

In this paper we obtain bounds for the decay rate for solutions to the nonlocal problem $\partial_t u(t,x) = \int_{\R^n} J(x,y)[u(t,y) - u(t,x)] dy$. Here we deal with bounded kernels $J$ but with polynomial tails, that is, we assume a lower bound of the form $J(x,y) \geq c_1|x-y|^{-(n + 2σ)}$, for $|x - y| > c_2$. Our estimates takes the form $\|u(t)\|_{L^q(\R^n)} \leq C t^{-\frac{n}{2σ} (1 - \frac{1}{q})}$ for $t$ large.

math.AP↗

arXiv2013

A nonlocal two phase Stefan problem

We study a nonlocal version of the two-phase Stefan problem, which models a phase transition problem between two distinct phases evolving to distinct heat equations. Mathematically speaking, this consists in deriving a theory for sign-changing solutions of the equation, ut = J * v - v, v = Γ(u), where the monotone graph is given by Γ(s) = sign(s)(|s|-1)+ . We give general results of existence, uniqueness and comparison, in the spirit of [2]. Then we focus on the study of the asymptotic behaviour for sign-changing solutions, which present challenging difficulties due to the non-monotone evolution of each phase.

math.AP↗

arXiv2012

Large Time Behavior of Periodic Viscosity Solutions for Uniformly Elliptic Integro-Differential Equations

In this paper, we study the large time behavior of solutions of a class of parabolic fully nonlinear integro-differential equations in a periodic setting. In order to do so, we first solve the ergodic problem}(or cell problem), i.e. we construct solutions of the form $λt + v(x)$. We then prove that solutions of the Cauchy problem look like those specific solutions as time goes to infinity. We face two key difficulties to carry out this classical program: (i) the fact that we handle the case of "mixed operators" for which the required ellipticity comes from a combination of the properties of the local and nonlocal terms and (ii) the treatment of the superlinear case (in the gradient variable). Lipschitz estimates previously proved by the authors (2012) and Strong Maximum principles proved by the third author (2012) play a crucial role in the analysis.

math.AP↗

arXiv2012

A Bellman approach for two-domains optimal control problems in $\R^N$

This article is the starting point of a series of works whose aim is the study of deterministic control problems where the dynamic and the running cost can be completely different in two (or more) complementary domains of the space $\R^N$. As a consequence, the dynamic and running cost present discontinuities at the boundary of these domains and this is the main difficulty of this type of problems. We address these questions by using a Bellman approach: our aim is to investigate how to define properly the value function(s), to deduce what is (are) the right Bellman Equation(s) associated to this problem (in particular what are the conditions on the set where the dynamic and running cost are discontinuous) and to study the uniqueness properties for this Bellman equation. In this work, we provide rather complete answers to these questions in the case of a simple geometry, namely when we only consider two different domains which are half spaces: we properly define the control problem, identify the different conditions on the hyperplane where the dynamic and the running cost are discontinuous and discuss the uniqueness properties of the Bellman problem by either providing explicitly the minimal and maximal solution or by giving explicit conditions to have uniqueness.

math.AP↗

arXiv2012

Lipschitz Regularity of Solutions for Mixed Integro-Differential Equations

We establish new Hoelder and Lipschitz estimates for viscosity solutions of a large class of elliptic and parabolic nonlinear integro-differential equations, by the classical Ishii-Lions's method. We thus extend the Hoelder regularity results recently obtained by Barles, Chasseigne and Imbert (2011). In addition, we deal with a new class of nonlocal equations that we term mixed integro-differential equations. These equations are particularly interesting, as they are degenerate both in the local and nonlocal term, but their overall behavior is driven by the local-nonlocal interaction, e.g. the fractional diffusion may give the ellipticity in one direction and the classical diffusion in the complementary one.

math.AP↗