Nonlinear PDEs of mixed type arising in mechanics and geometry

March 17 to March 21, 2008

at the

American Institute of Mathematics, San Jose, California

organized by

Gui-Qiang Chen, Tai-Ping Liu, Richard Schoen, and Marshall Slemrod

Original Announcement

This workshop will be devoted to the study of nonlinear partial differential equations of mixed hyperbolic and elliptic type arising in conservation laws, continuum mechanics, differential geometry, relativity, and string theory.

In the past two years there has been a surge of activity on nonlinear partial differential equations of mixed type where the problems that had long lain dormant because of lack of new ideas have been successfully attacked. However, the problems that can be studied and should be studied are much more general that what has been done so far. For example, the simple act of crumpling a piece of paper can be modeled as a "transonic" problem which possesses all the intuition gained by everyday experience and all the difficulty of systems of conservation laws exhibiting change of type. The "transonic" nature comes from the basic theory of differential geometry where the original Gauss curvature of the flat paper is zero and hence must be preserved under the isometry of the crumpling by Gauss's "theorem egregium". The zero Gauss curvature translates into making the Gauss-Codazzi system being everywhere sonic. The sonic problem with a little work can then be made a special case of more general transonic problems. Of course, many more problems arise in geometry and mechanics. Some recent publications on isometric embedding of Riemannian manifolds in Euclidean spaces have more examples, but ironically paper crumpling or any problems arising from day to day mechanics is not on their list. This workshop provides a venue to trade problems, methods, and ideas and to form new contacts with mathematicians, engineers, physicists, who might not normally meet in their usual day to day schedule.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Isometric immersions and compensated compactness
by  Gui-Qiang Chen, Marshall Slemrod, and Dehua Wang