WorldCat Identities

Souganidis, P. E.

Overview
Works: 27 works in 46 publications in 1 language and 40 library holdings
Genres: Periodicals 
Roles: Author
Classifications: QA377, 515.35305
Publication Timeline
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Most widely held works by P. E Souganidis
Differential Games and Representation Formulas for Solutions of Hamilton-Jacobi-Isaacs Equations by L. C Evans( Book )

3 editions published in 1983 in English and held by 3 WorldCat member libraries worldwide

Recent work by the authors and others has demonstrated the connections between the dynamic programming approach for two-person, zero-sum differential games and the new notion of viscosity solutions of Hamilton-Jacobi PDE, (Partial Differential Equations). The basic idea is that the dynamic programming optimality conditions imply that the values of a two-person, zero-sum differential game are viscosity solutions of appropriate PDE. This paper proves the above, when the values of the differential games are defined following Elliott-Kalton. This results in a great simplification in the statements and proofs, as the definitions are explicit and do not entail any kind of approximations. Moreover, as an application of the above results, the paper contains a representation formula for the solution of a fully nonlinear first-order PDE. This is then used to prove results about the level sets of solutions of Hamilton-Jacobi equations with homogeneous Hamiltonians. These results are also related to the theory of Huygen's principle and geometric optics
Viscosity solutions of second-order equations, stochastic control and stochastic differential games by P. L Lions( Book )

3 editions published between 1986 and 1987 in English and held by 3 WorldCat member libraries worldwide

Numerical analysis : convergence of MUSCL type method for scalar conservation laws = Analyse numérique by P. L Lions( Book )

2 editions published in 1991 in English and held by 2 WorldCat member libraries worldwide

Wavefront Propagation for Reaction-Diffusion Systems of PDE by Guy Barles( Book )

3 editions published in 1989 in English and held by 2 WorldCat member libraries worldwide

The theory of viscosity solutions for Hamilton-Jacobi equations is used to study the asymptotic behavior of solutions to certain systems of reaction-diffusion PDE. Our principal result characterizes the region of convergence of the solution to an unstable rest point as the set where the solution of an appropriate Hamilton-Jacobi equation is positive. Keywords include: Partial differential equations; Wavefront propagation
Uniqueness and singularities of cylindrically symmetric surfaces moving by mean curvature by H. Mete Soner( Book )

2 editions published in 1991 in English and held by 2 WorldCat member libraries worldwide

Abstract: "We consider the evolution of rotationally symmetric surfaces under mean curvature flow. We obtain local information around the geometric singularities and then prove uniqueness of weak solutions for a large class of initial data. The torus and dumbell are included in this class."
Phase transitions and generalized motion by mean curvature by L. C Evans( Book )

2 editions published between 1990 and 1991 in English and held by 2 WorldCat member libraries worldwide

Abstract: "We study the limiting behavior of solutions to appropriately rescaled versions of the Allen-Cahn equation, a model for phase transition in polycrystalline material. We rigourously establish the existence in the limit of a phase-antiphase interface evolving according to mean curvature motion. This assertion is valid for all positive time, the motion interpreted in the generalized sense of Evans-Spruck and Chen-Giga-Goto after the onset of geometric singularities."
Convergence of approximation schemes for fully nonlinear second order equations by Guy Barles( Book )

2 editions published in 1990 in English and held by 2 WorldCat member libraries worldwide

Differential Games, Optimal Control and Directional Derivatives of Viscosity Solutions of Bellman's and Isaacs' Equations by P. L Lions( Book )

2 editions published in 1984 in English and held by 2 WorldCat member libraries worldwide

Recent work by the authors and others has demonstrated the connections between the dynamic programming approach to optimal control theory and to two-person, zero-sum differential games problems and the new notion of viscosity solutions of Hamilton-Jacobi PDE's introduced by M.G. Crandall and P.L. Lions. In particular, it has been proved that the dynamic programming principle implies that the value function is the viscosity solution of the associated Hamilton-Jacobi-Bellman and Isaacs equations. In the present work, it is shown that viscosity super- and subsolutions of these equations must satisfy some inequalities called super- and subdynamic programming principle respectively. This is then used to prove the equivalence between the notion of viscosity solutions and the conditions, introduced by A. Subbotin, concerning the sign of certain generalized directional derivatives. (Author)
Front propagation for reaction-diffusion equations of bistable type by Guy Barles( Book )

2 editions published in 1990 in English and held by 2 WorldCat member libraries worldwide

Convergence of Difference Approximations of Quasilinear Evolution Equations by Michael G Crandall( Book )

2 editions published in 1984 in English and held by 2 WorldCat member libraries worldwide

The very successful theory of quasilinear evolution equations, which applies to many problems of mathematical physics, has been developed by T. Kato. The theory obtains solutions of quasilinear problems via contraction mappings which are defined by means of a theory of linear evolution equations also developed by Kato. In the current work we show how the existence and continuous dependence theorems obtained by Kato can be proved by discretization in time. As opposed to earlier work in this direction, the current results are much sharper concerning the continuity properties of the solutions of the discretized problem and the strength of the norms in which they converge
Differential games and directional derivatives of viscosity solutions of Isaacs' equations II by P. L Lions( Book )

3 editions published in 1985 in English and held by 2 WorldCat member libraries worldwide

Approximation Schemes for Viscosity Solutions of Hamilton-Jacobi Equations by Mathematics Research Center (United States. Army)( Book )

1 edition published in 1983 in English and held by 1 WorldCat member library worldwide

Equations of Hamilton-Jacobi type arise in many areas of application, including the calculus of variations of variations, control theory and differential games. Recently Crandall and Lions established the correct notion of generalized solutions for these equations. This article discusses the convergence of general approximation schemes to this solution and gives, under certain hypotheses, explicit error estimates. These results are then applied to obtain various representations. These include max-min representations of solutions relevant to the theory of differential games (which imply the existence of the value of the game), representations as limits of solutions of general explicit and implicit finite difference schemes, and as limits of several types of Trotter products. (Author)
A uniqueness result for viscosity solutions of second-order fully nonlinear partial differential equations by Rolf Jensen( Book )

2 editions published in 1987 in English and held by 1 WorldCat member library worldwide

Development in the theory of nonlinear first-order partial differential equations by Michael G Crandall( Book )

1 edition published in 1983 in English and held by 1 WorldCat member library worldwide

PDE-viscosity solution approach to some problems of large deviations by Wendell H Fleming( Book )

1 edition published in 1984 in English and held by 1 WorldCat member library worldwide

A limiting case for velocity averaging by B Perthame( Book )

1 edition published in 1998 in English and held by 1 WorldCat member library worldwide

Developments in the Theory of Nonlinear First-Order Partial Differential Equations by United States( Book )

1 edition published in 1983 in English and held by 1 WorldCat member library worldwide

The theory of scalar first-order fully nonlinear partial differential equations has recently enjoyed a strong development. One major step was a proof by M. Crandall and P.-L. Lions of the uniqueness of certain generalized solutions-called 'viscosity solutions'- of problems involving such equations with the scope to accommodate applications to, for example, differential games. Following this event there has been a continuous stream of work concerning the existence, approximation and representation of viscosity solutions of Hamilton-Jacobi equations as well as interaction of the theory of viscosity solutions and areas of application (primarily control theory and differential games), and refined uniqueness results. This survey paper, which corresponds to an invited address by the first author at an international symposium on differential equations held in March 1983 at the University of Alabama-Birmingham, introduces the relevant concepts and describes the major results up to, roughly, July 1983. (Author)
A PDE approach to geometric optics for certain semilinear parabolic equations by L. C Evans( Book )

2 editions published in 1987 in English and held by 1 WorldCat member library worldwide

A stochastic control approach to some large deviations problems by Wendell H Fleming( Book )

1 edition published in 1984 in English and held by 1 WorldCat member library worldwide

Communications in partial differential equations( )

in Undetermined and held by 1 WorldCat member library worldwide

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Audience level: 0.81 (from 0.62 for Communicat ... to 0.88 for PDE-viscos ...)

Languages
English (36)