Results 1 
9 of
9
The infinity Laplacian with a transport term
 J. MATH. ANAL. APPL
, 2013
"... We consider the following problem: given a bounded domain Ω ⊂ Rn and a vector field ζ: Ω → Rn, find a solution to −∆∞u − 〈Du, ζ 〉 = 0 in Ω, u = f on ∂Ω, where ∆ ∞ is the 1−homogeneous infinity Laplace operator that is formally given by ∆∞u = 〈D2u DuDu  , DuDu  〉 and f a Lipschitz boundary datu ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
We consider the following problem: given a bounded domain Ω ⊂ Rn and a vector field ζ: Ω → Rn, find a solution to −∆∞u − 〈Du, ζ 〉 = 0 in Ω, u = f on ∂Ω, where ∆ ∞ is the 1−homogeneous infinity Laplace operator that is formally given by ∆∞u = 〈D2u DuDu  , DuDu  〉 and f a Lipschitz boundary datum. If we assume that ζ is a continuous gradient vector field then we obtain existence and uniqueness of a viscosity solution by an Lpapproximation procedure. Also we prove the stability of the unique solution with respect to ζ. In addition when ζ is more regular (Lipschitz continuous) but not necessarily a gradient, using tugofwar games we prove that this problem has a solution.
Finite Difference Methods for the infinity Laplace and pLaplace equations
"... Abstract. We build convergent discretizations and semiimplicit solvers for the Infinity Laplacian and the game theoretical pLaplacian. The discretizations simplify and generalize earlier ones. We prove convergence of the solution of the Wide Stencil finite difference schemes to the unique viscosi ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We build convergent discretizations and semiimplicit solvers for the Infinity Laplacian and the game theoretical pLaplacian. The discretizations simplify and generalize earlier ones. We prove convergence of the solution of the Wide Stencil finite difference schemes to the unique viscosity solution of the underlying equation. We build a semiimplicit solver, which solves the Laplace equation as each step. It is fast in the sense that the number of iterations is independent of the problem size. This is an improvement over previous explicit solvers, which are slow due to the CFLcondition. 1.
A GameTree approach to discrete infinity Laplacian with running costs
, 2013
"... We give a selfcontained and elementary proof for boundedness, existence, and uniqueness of solutions to dynamic programming principles (DPP) for biased tugofwar games with running costs. The domain we work in is very general, and as a special case contains metric spaces. Technically, we introduc ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
We give a selfcontained and elementary proof for boundedness, existence, and uniqueness of solutions to dynamic programming principles (DPP) for biased tugofwar games with running costs. The domain we work in is very general, and as a special case contains metric spaces. Technically, we introduce gametrees and show that a discretized flow converges uniformly, from which we obtain not only the existence, but also the uniqueness. Our arguments are entirely deterministic, and also do not rely on (semi)continuity in any way; in particular, we do not need to mollify the DPP at the boundary for wellposedness.
AN OBSTACLE PROBLEM FOR TUGOFWAR GAMES
"... We consider the obstacle problem for the infinity Laplace equation. Given a Lipschitz boundary function and a Lipschitz obstacle we prove the existence and uniqueness of a super infinityharmonic function constrained to lie above the obstacle which is infinity harmonic where it lies strictly above t ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
We consider the obstacle problem for the infinity Laplace equation. Given a Lipschitz boundary function and a Lipschitz obstacle we prove the existence and uniqueness of a super infinityharmonic function constrained to lie above the obstacle which is infinity harmonic where it lies strictly above the obstacle. Moreover, we show that this function is the limit of value functions of a game we call obstacle tugofwar.
A GAME INTERPRETATION OF THE NEUMANN PROBLEM FOR FULLY NONLINEAR PARABOLIC AND ELLIPTIC EQUATIONS
"... Abstract. We provide a deterministiccontrolbased 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 twoperson games depending on a small parameter ε which extend those proposed by ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We provide a deterministiccontrolbased 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 twoperson 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 of the PDE as ε tends to zero. Moreover, our construction allows us to treat both the oblique and the mixed type DirichletNeumann boundary conditions. hal00670551, version 2 13 Dec 2012 1.
Nonlinear elliptic partial differential equations and pharmonic functions on graphs
, 2013
"... In this article we study the wellposedness (uniqueness and existence of solutions) of nonlinear elliptic Partial Differential Equations (PDEs) on a finite graph. These results are obtained using the discrete comparison principle and connectivity properties of the graph. This work is in the spirit ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
In this article we study the wellposedness (uniqueness and existence of solutions) of nonlinear elliptic Partial Differential Equations (PDEs) on a finite graph. These results are obtained using the discrete comparison principle and connectivity properties of the graph. This work is in the spirit of the theory of viscosity solutions for partial differential equations. The equations include the graph Laplacian, the pLaplacian, the Infinity Laplacian, and the Eikonal operator on the graph.