LEADER 08796nam 22006855 450 001 996466643603316 005 20200702234133.0 010 $a3-319-54208-7 024 7 $a10.1007/978-3-319-54208-9 035 $a(CKB)4340000000062307 035 $a(DE-He213)978-3-319-54208-9 035 $a(MiAaPQ)EBC6301220 035 $a(MiAaPQ)EBC5595830 035 $a(Au-PeEL)EBL5595830 035 $a(OCoLC)990345717 035 $a(PPN)202990281 035 $a(EXLCZ)994340000000062307 100 $a20170615d2017 u| 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aDynamical and Geometric Aspects of Hamilton-Jacobi and Linearized Monge-Ampère Equations$b[electronic resource] $eVIASM 2016 /$fby Nam Q. Le, Hiroyoshi Mitake, Hung V. Tran ; edited by Hiroyoshi Mitake, Hung V. Tran 205 $a1st ed. 2017. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2017. 215 $a1 online resource (VII, 228 p. 16 illus., 1 illus. in color.) 225 1 $aLecture Notes in Mathematics,$x0075-8434 ;$v2183 311 $a3-319-54207-9 320 $aIncludes bibliographical references. 327 $aIntro -- Preface -- Contents -- Part I The Second Boundary Value Problem of the Prescribed Affine Mean Curvature Equation and Related Linearized Monge-Ampe?re 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-Ampe?re and Linearized Monge-Ampe?re Equations -- 1.3 Proof of Global W4,p and C4,? 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-Ampe?re Equation -- 2.1 The Linearized Monge-Ampe?re Equation and Interior Regularity of Its Solution -- 2.1.1 The Linearized Monge-Ampe?re Equation -- 2.1.2 Linearized Monge-Ampe?re 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-Ampe?re Equation -- 2.2 Interior Harnack and Ho?lder Estimates for Linearized Monge-Ampe?re -- 2.2.1 Proof of Caffarelli-Gutie?rrez's Harnack Inequality -- 2.2.2 Proof of the Interior Ho?lder Estimates for the Inhomogeneous Linearized Monge-Ampe?re Equation -- 2.3 Global Ho?lder Estimates for the Linearized Monge-Ampe?re Equations. 327 $a2.3.1 Boundary Ho?lder Continuity for Solutions of Non-uniformly Elliptic Equations -- 2.3.2 Savin's Localization Theorem -- 2.3.3 Proof of Global Ho?lder Estimates for the Linearized Monge-Ampe?re Equation -- References -- 3 The Monge-Ampe?re Equation -- 3.1 Maximum Principles and Sections of the Monge-Ampe?re Equation -- 3.1.1 Basic Definitions -- 3.1.2 Examples and Properties of the Normal Mapping and the Monge-Ampe?re 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-Ampe?re Equation -- 3.2.1 Compactness of Solutions to the Monge-Ampe?re Equation -- 3.2.2 Caffarelli's Localization Theorem -- 3.2.3 Strict Convexity and C1,? 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. 327 $a6 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. 330 $aConsisting of two parts, the first part of this volume is an essentially self-contained exposition of the geometric aspects of local and global regularity theory for the Monge?Ampère and linearized Monge?Ampère equations. As an application, we solve the second boundary value problem of the prescribed affine mean curvature equation, which can be viewed as a coupling of the latter two equations. Of interest in its own right, the linearized Monge?Ampère equation also has deep connections and applications in analysis, fluid mechanics and geometry, including the semi-geostrophic equations in atmospheric flows, the affine maximal surface equation in affine geometry and the problem of finding Kahler metrics of constant scalar curvature in complex geometry. Among other topics, the second part provides a thorough exposition of the large time behavior and discounted approximation of Hamilton?Jacobi equations, which have received much attention in the last two decades, and a new approach to the subject, the nonlinear adjoint method, is introduced. The appendix offers a short introduction to the theory of viscosity solutions of first-order Hamilton?Jacobi equations. . 410 0$aLecture Notes in Mathematics,$x0075-8434 ;$v2183 606 $aPartial differential equations 606 $aCalculus of variations 606 $aDifferential geometry 606 $aPartial Differential Equations$3https://scigraph.springernature.com/ontologies/product-market-codes/M12155 606 $aCalculus of Variations and Optimal Control; Optimization$3https://scigraph.springernature.com/ontologies/product-market-codes/M26016 606 $aDifferential Geometry$3https://scigraph.springernature.com/ontologies/product-market-codes/M21022 615 0$aPartial differential equations. 615 0$aCalculus of variations. 615 0$aDifferential geometry. 615 14$aPartial Differential Equations. 615 24$aCalculus of Variations and Optimal Control; Optimization. 615 24$aDifferential Geometry. 676 $a515.353 700 $aLe$b Nam Q$4aut$4http://id.loc.gov/vocabulary/relators/aut$0742222 702 $aMitake$b Hiroyoshi$4aut$4http://id.loc.gov/vocabulary/relators/aut 702 $aTran$b Hung V$4aut$4http://id.loc.gov/vocabulary/relators/aut 702 $aMitake$b Hiroyoshi$4edt$4http://id.loc.gov/vocabulary/relators/edt 702 $aTran$b Hung V$4edt$4http://id.loc.gov/vocabulary/relators/edt 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a996466643603316 996 $aDynamical and Geometric Aspects of Hamilton-Jacobi and Linearized Monge-Ampère Equations$92004352 997 $aUNISA