Self-dual Partial Differential Systems and Their Variational Principles

by Ghoussoub, Nassif

• ISBN13: 9780387848969
• ISBN10: 0387848967
• Format: Hardcover
• Copyright: 2009-01-02
• Publisher: Springer Nature

Summary

This text is intended for a beginning graduate course on convexity methods for PDEs. The generality chosen by the author puts this under the classification of 'œfunctional analysis'. The applications, however, require a fair knowledge of classical analysis and PDEs which is needed to make judicious choices of function spaces where the self-dual variational principles need to be applied, and these choices necessarily require prior knowledge of the expected regularity of the (weak) solutions. While this text contains many new results, it is the author's hope that this material will soon become standard for all graduate students.

Table of Contents

Preface.- Introduction.- Legendre-Fenchel Duality on Phase Space.- Self-dual Lagrangians on Phase Space.- Skew-adjoint Operators and Self-dual Lagrangians.- Self-dual Vector Fields and Their Calculus.- Variational Principles for Completely Self-dual Functionals.- Semigroups of Contractions Associated to Self-dual Lagrangians.- Iteration of Self-dual Lagrangians and Multiparameter Evolutions.- Direct Sum of Completely Self-dual Functionals.- Semilinear Evolution with Self-dual Boundary Conditions.- The Class of Antisymmetric Hamiltonians.- Variational Principles for Self-dual Functionals and First Applications.- The Role of the Co-Hamiltonian in Self-dual Variational Problems.- Direct Sum of Self-dual Functionals and Hamiltonian Systems.- Superposition of Interacting Self-dual Functionals.- Hamiltonian Systems of Partial Differential Equations.- The Self-dual Palais-Smale Condition for Noncoercive Functionals.- Navier-Stokes and other Self-dual Nonlinear Evolutions.- References.