Dynamical and geometric aspects of Hamilton-Jacobi and linearized Monge-Ampère equations : VIASM 2016 /

Dynamical and geometric aspects of Hamilton-Jacobi and linearized Monge-Ampère equations : VIASM 2016 /

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Bibliographic Details
Author / Creator:Le, Nam Q., author.
Imprint:Cham, Switzerland : Springer, 2017.
Description:1 online resource (vii, 228 pages) : illustrations (some color)
Language:English
Series:Lecture notes in mathematics, 0075-8434 ; 2183
Lecture notes in mathematics (Springer-Verlag) ; 2183.
Subject:
Hamilton-Jacobi equations -- Congresses.
Monge-Ampère equations -- Congresses.
Hamilton-Jacobi equations.
Monge-Ampère equations.
Conference papers and proceedings.
Format: E-Resource Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/11307880
Hidden Bibliographic Details
Other authors / contributors:Mitake, Hiroyoshi, author editor.
Tran, Hung V., author editor.
ISBN:9783319542089
3319542087
3319542079
9783319542072
9783319542072
Digital file characteristics:PDF
text file
Notes:Online resource; title from PDF title page (SpringerLink, viewed June 21, 2017).
Summary: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.
Other form:Printed edition: 9783319542072
Standard no.:10.1007/978-3-319-54208-9
Table of Contents:
  • 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
  • 2.1 The Linearized Monge-Ampère Equation and Interior Regularity of Its Solution2.1.1 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 Equations2.3.1 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

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