Systems of Nonlinear Partial Differential Equations

This volume contains the proceedings of a NATO/London Mathematical Society Advanced Study Institute held in Oxford from 25 July - 7 August 1982. The institute concerned the theory and applications of systems of nonlinear partial differential equations, with emphasis on techniques appropriate to systems of more than one equation. Most of the lecturers and participants were analysts specializing in partial differential equations, but also present were a number of numerical analysts, workers in mechanics, and other applied mathematicians. The organizing committee for the institute was J.M. Ball (Heriot-Watt), T.B. Benjamin (Oxford), J. Carr (Heriot-Watt), C.M. Dafermos (Brown), S. Hildebrandt (Bonn) and J.S. pym (Sheffield) . The programme of the institute consisted of a number of courses of expository lectures, together with special sessions on different topics. It is a pleasure to thank all the lecturers for the care they took in the preparation of their talks, and S.S. Antman, A.J. Chorin, J.K. Hale and J.E. Marsden for the organization of their special sessions. The institute was made possible by financial support from NATO, the London Mathematical Society, the u.S. Army Research Office, the u.S. Army European Research Office, and the u.S. National Science Foundation. The lectures were held in the Mathematical Institute of the University of Oxford, and residential accommodation was provided at Hertford College.

Table of Contents:
I Expository Lectures.- Algebraic and Topological Invariants for Reaction-Diffusion Equations.- Hyperbolic Systems of Conservation Laws.- Ill-Posed Problems in Thermoelasticity Theory Lecture 1 Twinning of Thermoelastic Materials.- Lecture 2 Problems for Infinite Elastic Prisms.- Lecture 3 St.-Venantr–s Problem for Elastic Prisms.- Nonlinear Systems in Optimal Control Theory and Related Topics.- The Regularity Problem of Extremals of Variational Integrals.- Some Aspects of the Regularity Theory for Nonlinear Elliptic Systems.- Quasilinear Elliptic Systems in Diagonal Form.- Topics in Bifurcation Theory Lecture 1 A Brief Introduction to Bifurcation Theory.- Lecture 2 Unfoldings.- Lecture 3 Symmetry in Bifurcation Theory.- The Compensated Compactness Method Applied to Systems of Conservation Laws.- II Special Sessions.- a. Problems in Nonlinear Elasticity, organized by S. S. Antman (University of Maryland).- Coercivity Conditions in Nonlinear Elasticity.- Constitutive Inequalities and Dynamic Stability in the Linear Theories of Elasticity, Thermoelasticity and Viscoelasticity.- Generalized Solutions to Conservation Laws.- Stability of the Elastica.- Group Theoretic Classification of Conservation Laws in Elasticity.- b. Applications of Bifurcation Theory to Mechanics, organized by J. E. Marsden (University of California, Berkeley).- Phase Transitions Via Bifurcation from Heteroclinic Orbits.- Bifurcation under Continuous Groups of Symmetries.- Morse Decompositions and Global Continuation of Periodic Solutions for Singularly Perturbed Delay Equations.- Bifurcation and Linearization Stability in the Traction Problem.- Singular Elliptic Eigenvalue Problems for Equations and Systems.- c. Nonelliptic Problems and Phase Transitions, organized by J. M. Ball (Heriot-WattUniversity).- Regularization of Non Elliptic Variational Problems.- Remarks on the Relaxation of Integrals of the Calculus of Variations.- A Diffusion Equation with a Nonmonovone Constitutive Function.- An Admissibility Criterion for Fluids Exhibiting Phase Transitions.- d. Dynamical Systems and Partial Differential Equations, organized by J. K. Hale (Brown University).- Stabilization of Solutions for a System with a Continuum of Equilibria and Distinct Diffusion Coefficients.- Relation between the Trapped Rays and the Distribution of the Eigenfrequencies of the Wave Equation in the Exterior of an Obstacle.- Stabilization Properties for Nonlinear Degenerate Parabolic Equations with Cut-Off Diffusivity.- Dynamics in Parabolic Equations — An Example.