All a b c d e f g h i j k l m n o p q r s t u v w x y z Other

Page 1 Next

Displaying 1 – 20 of 24

Showing per page

A game interpretation of the Neumann problem for fully nonlinear parabolic and elliptic equations

Jean-Paul Daniel (2013)

ESAIM: Control, Optimisation and Calculus of Variations

We provide a deterministic-control-based interpretation for a broad class of fully nonlinear parabolic and elliptic PDEs with continuous Neumann boundary conditions in a smooth domain. We construct families of two-person games depending on a small parameter ε which extend those proposed by Kohn and Serfaty [21]. These new games treat a Neumann boundary condition by introducing some specific rules near the boundary. We show that the value function converges, in the viscosity sense, to the solution...

A general Hamilton-Jacobi framework for non-linear state-constrained control problems

Albert Altarovici, Olivier Bokanowski, Hasnaa Zidani (2013)

ESAIM: Control, Optimisation and Calculus of Variations

The paper deals with deterministic optimal control problems with state constraints and non-linear dynamics. It is known for such problems that the value function is in general discontinuous and its characterization by means of a Hamilton-Jacobi equation requires some controllability assumptions involving the dynamics and the set of state constraints. Here, we first adopt the viability point of view and look at the value function as its epigraph. Then, we prove that this epigraph can always be described...

A Hamilton-Jacobi approach to junction problems and application to traffic flows

Cyril Imbert, Régis Monneau, Hasnaa Zidani (2013)

ESAIM: Control, Optimisation and Calculus of Variations

This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a “junction”, that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They...

A Hölder infinity Laplacian

Antonin Chambolle, Erik Lindgren, Régis Monneau (2012)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we study the limit as p → ∞ of minimizers of the fractional Ws,p-norms. In particular, we prove that the limit satisfies a non-local and non-linear equation. We also prove the existence and uniqueness of solutions of the equation. Furthermore, we prove the existence of solutions in general for the corresponding inhomogeneous equation. By making strong use of the barriers in this construction, we obtain some regularity results.

A Hölder infinity Laplacian

Antonin Chambolle, Erik Lindgren, Régis Monneau (2012)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we study the limit as p → ∞ of minimizers of the fractional Ws,p-norms. In particular, we prove that the limit satisfies a non-local and non-linear equation. We also prove the existence and uniqueness of solutions of the equation. Furthermore, we prove the existence of solutions in general for the corresponding inhomogeneous equation. By making strong use of the barriers in this construction, we obtain some regularity results.

Exponential convergence for a convexifying equation

Guillaume Carlier, Alfred Galichon (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We consider an evolution equation similar to that introduced by Vese in [Comm. Partial Diff. Eq. 24 (1999) 1573–1591] and whose solution converges in large time to the convex envelope of the initial datum. We give a stochastic control representation for the solution from which we deduce, under quite general assumptions that the convergence in the Lipschitz norm is in fact exponential in time.

Exponential convergence for a convexifying equation

Guillaume Carlier, Alfred Galichon (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We consider an evolution equation similar to that introduced by Vese in [Comm. Partial Diff. Eq. 24 (1999) 1573–1591] and whose solution converges in large time to the convex envelope of the initial datum. We give a stochastic control representation for the solution from which we deduce, under quite general assumptions that the convergence in the Lipschitz norm is in fact exponential in time.

Exponential convergence for a convexifying equation

Guillaume Carlier, Alfred Galichon (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We consider an evolution equation similar to that introduced by Vese in [Comm. Partial Diff. Eq. 24 (1999) 1573–1591] and whose solution converges in large time to the convex envelope of the initial datum. We give a stochastic control representation for the solution from which we deduce, under quite general assumptions that the convergence in the Lipschitz norm is in fact exponential in time.

Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations

Guy Barles, Emmanuel Chasseigne, Cyril Imbert (2011)

Journal of the European Mathematical Society

This paper is concerned with the Hölder regularity of viscosity solutions of second-order, fully non-linear elliptic integro-differential equations. Our results rely on two key ingredients: first we assume that, at each point of the domain, either the equation is strictly elliptic in the classical fully non-linear sense, or (and this is the most original part of our work) the equation is strictly elliptic in a non-local non-linear sense we make precise. Next we impose some regularity and growth...

Homogenization of a semilinear parabolic PDE with locally periodic coefficients: a probabilistic approach

Abdellatif Benchérif-Madani, Étienne Pardoux (2007)

ESAIM: Probability and Statistics

In this paper, a singular semi-linear parabolic PDE with locally periodic coefficients is homogenized. We substantially weaken previous assumptions on the coefficients. In particular, we prove new ergodic theorems. We show that in such a weak setting on the coefficients, the proper statement of the homogenization property concerns viscosity solutions, though we need a bounded Lipschitz terminal condition.

Subharmonic functions in sub-Riemannian settings

Andrea Bonfiglioli, Ermanno Lanconelli (2013)

Journal of the European Mathematical Society

In this paper we furnish mean value characterizations for subharmonic functions related to linear second order partial differential operators with nonnegative characteristic form, possessing a well-behaved fundamental solution Γ . These characterizations are based on suitable average operators on the level sets of Γ . Asymptotic characterizations are also considered, extending classical results of Blaschke, Privaloff, Radó, Beckenbach, Reade and Saks. We analyze as well the notion of subharmonic function...

The principal eigenvalue of the ∞-laplacian with the Neumann boundary condition

Stefania Patrizi (2011)

ESAIM: Control, Optimisation and Calculus of Variations

We prove the existence of a principal eigenvalue associated to the ∞-Laplacian plus lower order terms and the Neumann boundary condition in a bounded smooth domain. As an application we get uniqueness and existence results for the Neumann problem and a decay estimate for viscosity solutions of the Neumann evolution problem.

Currently displaying 1 – 20 of 24

Page 1 Next