A beginner's guide to the theory of viscosity solutions / Shigeaki Koike

Auteur: Koike, Shigeaki (1958-) - AuteurType de document: Livre numériqueCollection: MSJ memoirs ; 13Langue: anglaisÉditeur: Tokyo : Mathematical Society of Japan, 2014 ISBN: 9784931469280 Note: Publisher's description: "The notion of viscosity solutions was first introduced by M. G. Crandall and P.-L. Lions in 1981 to study first-order partial differential equations in nondivergence form, typically, Hamilton-Jacobi equations. Later, the study of viscosity solutions was extended to second-order elliptic/parabolic equations. It has turned out that viscosity solution theory is a powerful tool used by many researchers to investigate fully nonlinear second-order (degenerate) elliptic/parabolic equations arising in optimal control problems, differential games, mean curvature flow, phase transitions, mathematical finance, conservation laws, variational problems, etc. This text is an introduction to viscosity solution theory as indicated by the title. "After a brief history of weak solutions, it presents several uniqueness (comparison principle) and existence results, which are the main issues. For further topics, it chooses generalized boundary value problems and regularity results. In the Appendix, which is the hardest part, it provides proofs of several important propositions.'' Sujets MSC: 49L25 Calculus of variations and optimal control; optimization -- Hamilton-Jacobi theories, including dynamic programming -- Viscosity solutions
35F30 Partial differential equations -- General first-order equations and systems -- Boundary value problems for nonlinear first-order equations
35J25 Partial differential equations -- Elliptic equations and systems -- Boundary value problems for second-order elliptic equations
35J60 Partial differential equations -- Elliptic equations and systems -- Nonlinear elliptic equations
35J70 Partial differential equations -- Elliptic equations and systems -- Degenerate elliptic equations
En-ligne: OA - accès libre | Zentralblatt

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Publisher's description: "The notion of viscosity solutions was first introduced by M. G. Crandall and P.-L. Lions in 1981 to study first-order partial differential equations in nondivergence form, typically, Hamilton-Jacobi equations. Later, the study of viscosity solutions was extended to second-order elliptic/parabolic equations. It has turned out that viscosity solution theory is a powerful tool used by many researchers to investigate fully nonlinear second-order (degenerate) elliptic/parabolic equations arising in optimal control problems, differential games, mean curvature flow, phase transitions, mathematical finance, conservation laws, variational problems, etc. This text is an introduction to viscosity solution theory as indicated by the title.
"After a brief history of weak solutions, it presents several uniqueness (comparison principle) and existence results, which are the main issues. For further topics, it chooses generalized boundary value problems and regularity results. In the Appendix, which is the hardest part, it provides proofs of several important propositions.''

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