Viscosity solutions for monotone systems of second–order elliptic PDES

Ishii, Hitoshi; Koike, Shigeaki

doi:10.1080/03605309108820791

Cited by 121 publications

(158 citation statements)

References 12 publications

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“…V_(0) = V(0) is a simple consequence of (18). Let us prove, following [2], that F is a subsolution on the boundary.…”

Section: Asymptotic Analysis Of the Ergodic Measurementioning

confidence: 99%

Perturbed Dynamical Systems with an Attracting Singularity and Weak Viscosity Limits in Hamilton-Jacobi Equations

Perthame¹

1990

Transactions of the American Mathematical Society

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Abstract.We give a new PDE proof of the Wentzell-Freidlin theorem concerning small perturbations of a dynamical system ue = tp on dSl.We prove that, if b has a single attractive singular point, ue converges uniformly on compact subsets of Í2 , and with an exponential decay, to a constant p., and we determine p . We also treat the case of Neumann boundary condition. In order to do so, we perform the asymptotic analysis for some ergodic measure which leads to a study of the viscosity limit of a Hamilton-Jacobi equation. This is achieved under very general assumptions by using a weak formulation of the viscosity limits of these equations. Résumé.Nous donnons une nouvelle preuve, par des méthodes EDP, du théorème de Wentzell-Freidlin concernant les petites perturbations d'un système dynamique: Leue = - §Awe -b-Vue = 0 dans Q, ue = tp sur dSï.

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“…V_(0) = V(0) is a simple consequence of (18). Let us prove, following [2], that F is a subsolution on the boundary.…”

Section: Asymptotic Analysis Of the Ergodic Measurementioning

confidence: 99%

Perturbed Dynamical Systems with an Attracting Singularity and Weak Viscosity Limits in Hamilton-Jacobi Equations

Perthame¹

1990

Transactions of the American Mathematical Society

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“…Expanding its availability beyond scalar equations, viscosity solution theory for systems has been advanced to understand the optimal switching of controls for both deterministic [18,35], [41], [44] and stochastic [27,26,45] problems; these works offer a number of results on existence, uniqueness, and qualitative properties of solutions. On the other hand, the viscosity solution approach to nonlocal equations is still under development and is currently an active research area, cf.…”

Section: Introductionmentioning

confidence: 99%

Viscosity Solutions for a System of Integro-PDEs and Connections to Optimal Switching and Control of Jump-Diffusion Processes

2009

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Abstract. We develop a viscosity solution theory for a system of nonlinear degenerate parabolic integro-partial differential equations (IPDEs) related to stochastic optimal switching and control problems or stochastic games. In the case of stochastic optimal switching and control, we prove via dynamic programming methods that the value function is a viscosity solution of the IPDEs. In our setting the value functions or the solutions of the IPDEs are not smooth, so classical verification theorems do not apply.

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“…If problem (24) has no positive bounded solutions in R N −1 , then (22) has no nontrivial nonnegative bounded solutions in R N + which vanish on {x N = 0}. In the case when F 1 and F 2 are the Laplacian, this theorem appeared in [17].…”

Section: A Liouville Theorem In a Half-spacementioning

confidence: 95%

“…These operators are essential tools in control theory and in theory of large deviations, while general Isaac's operators are basic in game theory. We refer to [5], [24], and to the references in these papers, for a larger list of problems, where systems of type (1) appear. Next, we give the hypotheses we make on H k , which we suppose to be in the form (3).…”

Section: Under the Same Hypotheses System (2) (Without The Boundmentioning

confidence: 99%

Existence and non-existence results for fully nonlinear elliptic systems

Quaas¹,

Sirakov²

2009

Indiana Univ. Math. J.

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Abstract. We study systems of two elliptic equations, with right-hand sides with general power-like superlinear growth, and left-hand sides which are of Isaac's or Hamilton-Jacobi-Bellman type (however our results are new even for linear lefthand sides). We show that under appropriate growth conditions such systems have positive solutions in bounded domains, and that all such solutions are bounded in the uniform norm. We also get nonexistence results in unbounded domains.

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Viscosity solutions for monotone systems of second–order elliptic PDES

Cited by 121 publications

References 12 publications

Perturbed Dynamical Systems with an Attracting Singularity and Weak Viscosity Limits in Hamilton-Jacobi Equations

Perturbed Dynamical Systems with an Attracting Singularity and Weak Viscosity Limits in Hamilton-Jacobi Equations

Viscosity Solutions for a System of Integro-PDEs and Connections to Optimal Switching and Control of Jump-Diffusion Processes

Existence and non-existence results for fully nonlinear elliptic systems

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