Relaxation in Optimization Theory and Variational Calculus

Tomáš Roubiček

• Publisher: De Gruyter
• Year: 2011
• Print ISBN: 9783110145427
• eBook ISBN: 9783110811919
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

The series is devoted to the publication of high-level monographs which cover the whole spectrum of current nonlinear analysis and applications in various fields, such as optimization, control theory, systems theory, mechanics, engineering, and other sciences. One of its main objectives is to make available to the professional community expositions of results and foundations of methods that play an important role in both the theory and applications of nonlinear analysis. Contributions which are on the borderline of nonlinear analysis and related fields and which stimulate further research at the crossroads of these areas are particularly welcome.

Editor-in-Chief
Jürgen Appell, Würzburg, Germany

Honorary and Advisory Editors
Catherine Bandle, Basel, Switzerland
Alain Bensoussan, Richardson, Texas, USA
Avner Friedman, Columbus, Ohio, USA
Umberto Mosco, Worcester, Massachusetts, USA
Louis Nirenberg, New York, USA
Alfonso Vignoli, Rome, Italy

Editorial Board
Manuel del Pino, Santiago, Chile
Mikio Kato, Nagano, Japan
Wojciech Kryszewski, Toru?, Poland
Simeon Reich, Haifa, Israel

Please submit book proposals to Jürgen Appell.

Titles in planning include

Eduardo V. Teixeira, Free Boundary Problems: A Primer (2018)
Lucio Damascelli and Filomena Pacella, Morse Index of Solutions of Nonlinear Elliptic Equations (2019)
Rafael Ortega, Periodic Differential Equations in the Plane: A Topological Perspective (2019)
Cyril Tintarev, Profile Decompositions and Cocompactness: Functional-Analytic Theory of Concentration Compactness (2020)
Takashi Suzuki, Semilinear Elliptic Equations: Classical and Modern Theories (2021)

Table of contents

• Preface
• 1 Background generalities
• 1.1 Order and topology
• 1.2 Linear and convex analysis
• 1.3 Optimization theory
• 1.4 Function and measure spaces
• 1.5 Means of continuous functions
• 1.6 Some differential and integral equations
• 1.7 Non-cooperative game theory
• 2 Theory of convex compactifications
• 2.1 Convex compactifications
• 2.2 Canonical form of convex compactifications
• 2.3 Convex σ-compactifications
• 2.4 Approximation of convex compactifications
• 2.5 Extension of mappings
• 3 Young measures and their generalizations
• 3.1 Classical Young measures
• 3.2 Various generalizations
• 3.3 Convex compactifications of balls in Lp-spaces
• 3.4 Convex σ-compactifications of Lp-spaces
• 3.5 Approximation theory
• 3.6 Extensions of Nemytskiĭ mappings
• 4 Relaxation in optimization theory
• 4.1 Abstract optimization problems
• 4.2 Optimization problems on Lebesgue spaces
• 4.3 Example: Optimal control of dynamical systems
• 4.4 Example: Elliptic optimal control problems
• 4.5 Example: Parabolic optimal control problems
• 4.6 Example: Optimal control of integral equations
• 5 Relaxation in variational calculus I
• 5.1 Convex compactifications of Sobolev spaces
• 5.2 Relaxation of variational problems; p > 1
• 5.3 Optimality conditions for relaxed problems
• 5.4 Relaxation of variational problems; p= 1
• 5.5 Convex approximations of relaxed problems
• 6 Relaxation in variational calculus II
• 6.1 Prerequisities around quasiconvexity
• 6.2 Gradient generalized Young functionals
• 6.3 Relaxation scheme and its FEM-approximation
• 6.4 Further approximation: an inner case
• 6.5 Further approximation: an outer case
• 6.6 Double-well problem: sample calculations
• 7 Relaxation in game theory
• 7.1 Abstract game-theoretical problems
• 7.2 Games on Lebesgue spaces
• 7.3 Example: Games with dynamical systems
• 7.4 Example: Elliptic games
• Bibliography
• List of Symbols
• Index