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
.
Most widely held works by
P. E Souganidis
Differential Games and Representation Formulas for Solutions of HamiltonJacobiIsaacs 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 twoperson, zerosum differential games and the new notion of viscosity solutions of HamiltonJacobi PDE, (Partial Differential Equations). The basic idea is that the dynamic programming optimality conditions imply that the values of a twoperson, zerosum differential game are viscosity solutions of appropriate PDE. This paper proves the above, when the values of the differential games are defined following ElliottKalton. 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 firstorder PDE. This is then used to prove results about the level sets of solutions of HamiltonJacobi equations with homogeneous Hamiltonians. These results are also related to the theory of Huygen's principle and geometric optics
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 twoperson, zerosum differential games and the new notion of viscosity solutions of HamiltonJacobi PDE, (Partial Differential Equations). The basic idea is that the dynamic programming optimality conditions imply that the values of a twoperson, zerosum differential game are viscosity solutions of appropriate PDE. This paper proves the above, when the values of the differential games are defined following ElliottKalton. 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 firstorder PDE. This is then used to prove results about the level sets of solutions of HamiltonJacobi equations with homogeneous Hamiltonians. These results are also related to the theory of Huygen's principle and geometric optics
Viscosity solutions of secondorder 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
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
2 editions published in 1991 in English and held by 2 WorldCat member libraries worldwide
Wavefront Propagation for ReactionDiffusion 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 HamiltonJacobi equations is used to study the asymptotic behavior of solutions to certain systems of reactiondiffusion 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 HamiltonJacobi equation is positive. Keywords include: Partial differential equations; Wavefront propagation
3 editions published in 1989 in English and held by 2 WorldCat member libraries worldwide
The theory of viscosity solutions for HamiltonJacobi equations is used to study the asymptotic behavior of solutions to certain systems of reactiondiffusion 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 HamiltonJacobi 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."
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 AllenCahn equation, a model for phase transition in polycrystalline material. We rigourously establish the existence in the limit of a phaseantiphase interface evolving according to mean curvature motion. This assertion is valid for all positive time, the motion interpreted in the generalized sense of EvansSpruck and ChenGigaGoto after the onset of geometric singularities."
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 AllenCahn equation, a model for phase transition in polycrystalline material. We rigourously establish the existence in the limit of a phaseantiphase interface evolving according to mean curvature motion. This assertion is valid for all positive time, the motion interpreted in the generalized sense of EvansSpruck and ChenGigaGoto 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
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 twoperson, zerosum differential games problems and the new notion of viscosity solutions of HamiltonJacobi 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 HamiltonJacobiBellman 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)
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 twoperson, zerosum differential games problems and the new notion of viscosity solutions of HamiltonJacobi 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 HamiltonJacobiBellman 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 reactiondiffusion equations of bistable type by
Guy Barles(
Book
)
2 editions published in 1990 in English and held by 2 WorldCat member libraries worldwide
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
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
3 editions published in 1985 in English and held by 2 WorldCat member libraries worldwide
Approximation Schemes for Viscosity Solutions of HamiltonJacobi 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 HamiltonJacobi 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 maxmin 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)
1 edition published in 1983 in English and held by 1 WorldCat member library worldwide
Equations of HamiltonJacobi 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 maxmin 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 secondorder fully nonlinear partial differential equations by
Rolf Jensen(
Book
)
2 editions published in 1987 in English and held by 1 WorldCat member library worldwide
2 editions published in 1987 in English and held by 1 WorldCat member library worldwide
Development in the theory of nonlinear firstorder partial differential equations by
Michael G Crandall(
Book
)
1 edition published in 1983 in English and held by 1 WorldCat member library worldwide
1 edition published in 1983 in English and held by 1 WorldCat member library worldwide
PDEviscosity 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
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
1 edition published in 1998 in English and held by 1 WorldCat member library worldwide
Developments in the Theory of Nonlinear FirstOrder 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 firstorder 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 solutionscalled '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 HamiltonJacobi 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 AlabamaBirmingham, introduces the relevant concepts and describes the major results up to, roughly, July 1983. (Author)
1 edition published in 1983 in English and held by 1 WorldCat member library worldwide
The theory of scalar firstorder 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 solutionscalled '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 HamiltonJacobi 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 AlabamaBirmingham, 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
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
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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in Undetermined and held by 1 WorldCat member library worldwide
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Related Identities
 Lions, PierreLouis Author
 Evans, L. C. Author
 Barles, G. Author
 Crandall, M. G. Author
 Soner, H. M. Author
 WISCONSIN UNIVMADISON MATHEMATICS RESEARCH CENTER
 Fleming, W. H. Author
 University of Minnesota Institute for Mathematics and Its Applications
 Jensen, R. Author
 Bronsard, L.
Associated Subjects
Calculus of variations Conservation laws (Mathematics) Curvature Differential equations, NonlinearNumerical solutions Differential equations, Parabolic Differential equations, Partial Differential equations, PartialNumerical solutions Geometric measure theory HamiltonJacobi equations HamiltonJacobi equationsNumerical solutions Hypersurfaces Initial value problems Numerical analysis Phase transformations (Statistical physics) Reactiondiffusion equations Singularities (Mathematics) Viscosity solutions Wave equation
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