Dynamical and Geometric Aspects of Hamilton-Jacobi and Linearized Monge-Ampère Equations [[electronic resource] ] : VIASM 2016 / / by Nam Q. Le, Hiroyoshi Mitake, Hung V. Tran ; edited by Hiroyoshi Mitake, Hung V. Tran

• Autore: Le Nam Q
• Pubblicazione: Cham : , : Springer International Publishing : , : Imprint : Springer, , 2017
• Edizione: 1st ed. 2017.
• Descrizione fisica: 1 online resource (VII, 228 p. 16 illus., 1 illus. in color.)
• Disciplina: 515.353
• Soggetto topico: Partial differential equations; Calculus of variations; Differential geometry; Partial Differential Equations; Calculus of Variations and Optimal Control; Optimization; Differential Geometry
• Persona (resp. second.): MitakeHiroyoshi; TranHung V
• Nota di bibliografia: Includes bibliographical references.
• Nota di contenuto: Intro -- Preface -- Contents -- Part I The Second Boundary Value Problem of the Prescribed Affine Mean Curvature Equation and Related Linearized Monge-Ampère Equation -- Introduction -- Notation -- 1 The Affine Bernstein and Boundary Value Problems -- 1.1 The Affine Bernstein and Boundary Value Problems -- 1.1.1 Minimal Graph -- 1.1.2 Affine Maximal Graph -- 1.1.3 The Affine Bernstein Problem -- 1.1.4 Connection with the Constant Scalar Curvature Problem -- 1.1.5 The First Boundary Value Problem -- 1.1.6 The Second Boundary Value Problem of the Prescribed Affine Mean Curvature Equation -- 1.1.7 Solvability of the Second Boundary Value Problem -- 1.2 Existence of Solution to the Second Boundary Value Problem -- 1.2.1 Existence of Solution via Degree Theory and A Priori Estimates -- 1.2.2 Several Boundary Regularity Results for Monge-Ampère and Linearized Monge-Ampère Equations -- 1.3 Proof of Global W4,p and C4,α Estimates -- 1.3.1 Test Functions -- 1.3.2 L1 Bound and Lower Bound on the Hessian Determinant -- 1.3.3 Gradient Bound -- 1.3.4 Legendre Transform and Upper Bound on Hessian Determinant -- References -- 2 The Linearized Monge-Ampère Equation -- 2.1 The Linearized Monge-Ampère Equation and Interior Regularity of Its Solution -- 2.1.1 The Linearized Monge-Ampère Equation -- 2.1.2 Linearized Monge-Ampère Equations in Contexts -- 2.1.3 Difficulties and Expected Regularity -- 2.1.4 Affine Invariance Property -- 2.1.5 Krylov-Safonov's Harnack Inequality -- 2.1.6 Harnack Inequality for the Linearized Monge-Ampère Equation -- 2.2 Interior Harnack and Hölder Estimates for Linearized Monge-Ampère -- 2.2.1 Proof of Caffarelli-Gutiérrez's Harnack Inequality -- 2.2.2 Proof of the Interior Hölder Estimates for the Inhomogeneous Linearized Monge-Ampère Equation -- 2.3 Global Hölder Estimates for the Linearized Monge-Ampère Equations. ; 2.3.1 Boundary Hölder Continuity for Solutions of Non-uniformly Elliptic Equations -- 2.3.2 Savin's Localization Theorem -- 2.3.3 Proof of Global Hölder Estimates for the Linearized Monge-Ampère Equation -- References -- 3 The Monge-Ampère Equation -- 3.1 Maximum Principles and Sections of the Monge-Ampère Equation -- 3.1.1 Basic Definitions -- 3.1.2 Examples and Properties of the Normal Mapping and the Monge-Ampère Measure -- 3.1.3 Maximum Principles -- 3.1.4 John's Lemma -- 3.1.5 Comparison Principle and Applications -- 3.1.6 The Dirichlet Problem and Perron's Method -- 3.1.7 Sections of Convex Functions -- 3.2 Geometry of Sections of Solutions to the Monge-Ampère Equation -- 3.2.1 Compactness of Solutions to the Monge-Ampère Equation -- 3.2.2 Caffarelli's Localization Theorem -- 3.2.3 Strict Convexity and C1,α Estimates -- 3.2.4 Engulfing Property of Sections -- Appendix A: Auxiliary Lemmas -- Appendix B: A Heuristic Explanation of Trudinger-Wang's Non-smooth Example -- References -- Part II Dynamical Properties of Hamilton-Jacobi Equations via the Nonlinear Adjoint Method: Large Time Behavior and Discounted Approximation -- Introduction -- Notations -- References -- 4 Ergodic Problems for Hamilton-Jacobi Equations -- 4.1 Motivation -- 4.2 Existence of Solutions to Ergodic Problems -- References -- 5 Large Time Asymptotics of Hamilton-Jacobi Equations -- 5.1 A Brief Introduction -- 5.2 First-Order Case with Separable Hamiltonians -- 5.2.1 First Example -- 5.2.2 Second Example -- 5.3 First-Order Case with General Hamiltonians -- 5.3.1 Formal Calculation -- 5.3.2 Regularizing Process -- 5.3.3 Conservation of Energy and a Key Observation -- 5.3.4 Proof of Key Estimates -- 5.4 Degenerate Viscous Case -- 5.5 Asymptotic Profile of the First-Order Case -- 5.6 Viscous Case -- 5.7 Some Other Directions and Open Questions -- References. ; 6 Selection Problems in the Discounted Approximation Procedure -- 6.1 Selection Problems -- 6.1.1 Examples on Nonuniqueness of Ergodic Problems -- 6.1.2 Discounted Approximation -- 6.2 Regularizing Process -- 6.2.1 Regularizing Process and Construction of M -- 6.2.2 Stochastic Mather Measures -- 6.2.3 Key Estimates -- 6.3 Proof of Theorem 6.5 -- 6.4 Proof of the Commutation Lemma -- 6.5 Applications -- 6.5.1 Limit of u in Example 6.1 -- 6.5.2 Limit of u in Examples 6.3, 6.4 -- 6.6 Some Other Directions and Open Questions -- 6.6.1 Discounted Approximation Procedure -- 6.6.2 Vanishing Viscosity Procedure -- 6.6.3 Selection of Mather Measures -- References -- 7 Appendix of Part II -- 7.1 Motivation and Examples -- 7.1.1 Front Propagation Problems -- 7.1.2 Optimal Control Problems -- 7.1.2.1 Inviscid Cases -- 7.1.2.2 Viscous Cases -- 7.2 Definitions -- 7.3 Consistency -- 7.4 Comparison Principle and Uniqueness -- 7.5 Stability -- 7.6 Lipschitz Estimates -- 7.7 The Perron Method -- References.
• Sommario/riassunto: Consisting of two parts, the first part of this volume is an essentially self-contained exposition of the geometric aspects of local and global regularity theory for the Monge–Ampère and linearized Monge–Ampère equations. As an application, we solve the second boundary value problem of the prescribed affine mean curvature equation, which can be viewed as a coupling of the latter two equations. Of interest in its own right, the linearized Monge–Ampère equation also has deep connections and applications in analysis, fluid mechanics and geometry, including the semi-geostrophic equations in atmospheric flows, the affine maximal surface equation in affine geometry and the problem of finding Kahler metrics of constant scalar curvature in complex geometry. Among other topics, the second part provides a thorough exposition of the large time behavior and discounted approximation of Hamilton–Jacobi equations, which have received much attention in the last two decades, and a new approach to the subject, the nonlinear adjoint method, is introduced. The appendix offers a short introduction to the theory of viscosity solutions of first-order Hamilton–Jacobi equations. .
• Titolo autorizzato: Dynamical and Geometric Aspects of Hamilton-Jacobi and Linearized Monge-Ampère Equations
• ISBN: 3-319-54208-7
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 996466643603316
• Lo trovi qui: Univ. di Salerno
• Serie: Lecture Notes in Mathematics, . 0075-8434 ; ; 2183