PDE of Mixed Type: The Twin Challenges of Globalization and Diversity

Partial Differential Equations (PDE) of mixed elliptic-hyperbolic type arise in particular but interesting contexts such as transonic fluid flow and isometric embeddings of Riemannian manifolds whose curvature changes sign. Problems which involve mixed type PDE are difficult due in large measure to diversity; that is, the mixture of qualitative types competes with the fact that sharp PDE tools are often calibrated to the type of the equation. The most interesting and natural problems involve, of course, nonlinear equations, but progress on them remains inhibited due to a glaring lack of precise information on linear equations of mixed type. For example, even for linear equations, the question of what constitutes a well posed boundary value problem is particularly delicate as the question of desired regularity is crucial. Moreover, spectral theory is almost completely absent which complicates the treatment of natural problems which possess a variational structure with associated functionals which are strongly indefinite. For truly nonlinear problems, handling possible singularities or shocks is a main objective. As is typical for PDE problems, one often can reduce the question at hand to the presence of suitable a priori estimates. For mixed type equations, such estimates, even when locally available, need not be globalizable in a robust or clear-cut way. We will give a general overview of some of the interesting problems which involve mixed type PDE as well as some strategies for obtaining global information.