Colóquio 

21/09/2000, 16:30 — 17:30 — Anfiteatro Pa1, Pavilhão de Matemática
Vladimir Maz’ya, Linköpings Universitet

Some Open Problems in Partial Differential Equations

The development of the theory of elliptic boundary value problems in the last century is concentrated mainly around the 19th and 20th Hilbert problems, dealing with existence, analyticity, and regularity properties of solutions. It is believed that both problems have been principally solved.

I intend to show that Hilbert’s program is far from being exhausted and still is able to give strong impetus to the theory of elliptic equations. Among the deep problems which are waiting to be explored are:

1. Description of classes of elliptic systems as well as higher order elliptic equations and variational problems with analytic nonlinearities which have only analytic solutions.
2. Characterization of local and boundary singularities of nonlinear elliptic equations.
3. The study of boundary behaviour of solutions to nonlinear and higher order linear equations in domains with unrestricted boundaries. In particular, a circle of questions concentrated around Wiener’s criterion for regularity of a boundary point.