2004-2005 Milliman Lectures
Department of Mathematics
University of Washington

LUIS CAFFARELLI
University of Texas at Austin
(Sid Richardson Chair holder, Professor in Mathematics)
and the Texas Institute for Computations and Applied Mathematics

February 8, 9, and 10
4:00 - 5:00pm
Physics-Astronomy Building Room A118

Title of Lectures: Free Boundary Problems of Obstacle Type

Abstract:  Free boundary problems of obstacle type became of interest during the development of the theory of Variational Inequalities in the late 1960s, and applications surfaced in many areas of applied math, probability and geometry.

From the variational point of view, the problem consists of minimizing some variational integral , among those configurations staying above a given graph. From the probabilistic point of view, it concerns an optimal stopping time problem that gives rise to a Hamilton-Jacobi-Bellman equation.

Typical examples of the variational integrals involve those giving rise to second order linear or non linear equations (Laplacian, minimal surfaces, p-Laplacian), fourth order problems like the bi-Laplacian (clamped plates), or the fractional Laplacian for instance for boundary control problems or optimal stopping times for Levi processes. We will discuss matters of regularity, stability, and the geometric properties of the contact set and its free boundary.

Lecture I:  We will discuss what is a free boundary problem, in particular one of obstacle type, describe different areas, from geometry to fluid dynamics to probability in which they appear, and what type of information we seek.

Lecture II:  We will give a more detailed discussion of regularity and geometric behavior of solutions to these types of problems, as well as its interpretations, in particular in connection with recent work on non-local variational inequalities.

Lecture III:  We will discuss in greater detail the main techniques to tackle this type of problem, in particular "convexity" properties of solutions, and the role of monotonicity formulas.

Luis Caffarelli is a leader in the field of partial differential equations and their applications. He has been a professor at the University of Minnesota, the University of Chicago, the Courant Institute and the University of Texas. From 1986 to 1996 he was a permanent member at the Institute for Advanced Study in Princeton. In 1991 he was elected to the National Academy of Sciences. He has been awarded Doctor Honoris Causa from l'Ecole Normal Superieure, Paris; Universidad Autónoma de Madrid, and Universidad de la Plata, Argentina. He received the Bocher Prize in 1984.