Professor Luis Caffarelli received the 2005 Rolf Schock Prize in Mathematics. The prize diploma was conferred by Princess Christina of Sweden at a ceremony on October 27, 2005, in the Royal Swedish Academy of Music in Stockholm.

The presentation speech follows.

Princess Christina, President Gunnar, Your Excellencies, Distinguished Laureates, Ladies and Gentlemen,

Professor Luis A. Caffarelli receives this year's Schock Prize in Mathematics for his important contributions to the theory of nonlinear partial differential equations.

Luis Caffarelli was born in 1948 in Argentina. He obtained his Ph.D. in 1972 at the Universidad de Buenos Aires. (The tango was great, by the way.) In 1973 he moved to the United States, where he worked with Calixto P. Calderón in Minnesota. He has held professorships in Minnesota, at the Courant Institute in New York, at the University of Chicago, and at the Institute for Advanced Study in Princeton, New Jersey. Since 1997 he is a professor at the University of Texas at Austin. He is an honorary doctor of the École Normale Supérieure in Paris, the Universidad Autonoma de Madrid and the Universidad de la Plata. He has received a number of prizes, though none so far as big as the Schock Prize.

I studied theoretical philosophy at Stockholm University—then Stockholms högskola—a little before, and partly overlapping with, Rolf Schock. Never during my time there did I imagine that I would one day be trying to explain mathematical research that had been rewarded with his money. And yet this is exactly what I shall try to do now.

Calculus is a mathematical theory concerned with change. Its basic processes are differentiation and integration. I am well aware that these words do not belong to the general culture. But many particular instances of them do.

We speak about the speed of a car measured in meters per second or miles per hour. We speak about its acceleration, measured in meters per second per second. We speak about inflation in the economy, measured in percent per year. All these statements represent change, and they are instances of the mathematical concept of a derivative.

In the other direction, we sum our monthly salaries to an income per year; we talk about a country's gross national product. These are instances of the mathematical concept of integration.

Differential equations are the mathematician's foremost aid for describing change. Only in a completely static world would models built on these aids be unnecessary. In the simplest case, a process depends on one variable alone, for example time. It may then be described with the help of an ordinary differential equation. But more complex phenomena depend on several variables—perhaps time and, in addition, one, two or three space variables. Such processes require the use of partial differential equations and are very common in mathematical descriptions of natural phenomena.

The theory of these partial differential equations has a long history. However, results are best known when the equations are linear. This means, roughly speaking, that if we have two solutions, we may add them to get a third good solution. In the nonlinear case this is no longer so. Much less is known about such equations. Nevertheless, they constitute important models for many phenomena in the real world.

A classical problem which leads to a nonlinear equation is that of
moving materials: to move a pile of earth from one location to another
with as little effort as possible. The problem was posed by Gaspard
Monge (1746–1818) in his "Mémoire sur la théorie des déblais et
des remblais" (*Histoire de l'Académie Royale des Sciences de Paris
avec les Mémoires de Mathématiques et de Physique pour la même
année,* pp. 666–704, 1781). Here *déblai* means the
pile to be dug up and taken away, and *remblai* is the place to
which we shall take it—one senses the military applications of a
solution.

The problem was solved in Monge's original sense in 1976 by V. N. Sudakov. In 1999, C. Evans and W. Ganbo gave a different proof, and Caffarelli with coauthors Mikhail Feldman and Robert J. McCann gave a more constructive proof in 2002. This is an example of a nonlinear partial differential equation, the Monge–Ampère equation. (André Marie Ampère, 1775–1836, is mentioned in this context because he considered electric currents moving in a medium with varying resistance.) The Monge–Ampère equation has been the subject of many studies, among the most important being a series of articles by Caffarelli together with Louis Nirenberg and Joel Spruck, and, in the complex case, Joseph J. Kohn. This series has had a great importance for all subsequent work on the problem, both in real and complex analysis.

Usually important data serving to narrow down a solution are given at a boundary. If we are studying the weather, for instance, the earth's surface is such a boundary, where there are no vertical winds and where a certain exchange of heat takes place. During the time we are studying the weather, this surface can be considered to be fixed, and we can set the vertical wind to be zero.

All this will be totally different if we study thawing in the
frozen ground. Then there is a boundary between frozen and unfrozen
soil, but this boundary will vary; it cannot be considered to be
fixed. This is what we call a *free-boundary problem.* The
physical properties of a medium change when we go from ice to water or
from melted metal to metal in solid form.

For many years, since Newton and Leibniz, calculus has been concerned with gradual, smooth change. If a car moves along a road we can handle that and calculate its speed and acceleration. It moves perhaps an inch a millisecond.

But the gradual, smooth changes are not the only changes in nature. There are also sudden changes: cars sometimes collide; there are explosions; there are earthquakes; a metal can melt and there is a sharp, moving boundary between the solid and the melted metal. Classical calculus cannot handle this.

In a series of papers starting in 1990, Caffarelli introduced and studied so-called viscosity solutions to nonlinear partial differential equations, both the Monge–Ampère equation and the equation that models flow in a porous medium. This has proved to be an important means to get existence and uniqueness of solutions.

Caffarelli has worked with, among others, Avner Friedman, Louis Nirenberg, Joel Spruck, Hans Wilhelm Alt, Carlos E. Kenig, and Henrik Shahgholian at the Royal Institute of Technology.

We are happy to see among us here three of the prizewinner's colleagues, who lectured together with him yesterday at the symposium: Irene Gamba, Austin, Fang-Hua Lin of the Courant Institute, and Panagiotis (Takis) E. Souganidis, also of Austin.

For thirty years Luis Caffarelli has been the world's leading specialist in free-boundary problems for nonlinear partial differential equations. His pioneering methods have tackled many classical problems that have long defied mathematicians. It will probably be decades before scientists have utilized all the techniques he has created.

I cannot resist the temptation to share with you the
characterization of Luis Caffarelli that Takis Souganidis gave at
yesterday's symposium. He characterized our prize-winner as a
mathematician using two words: *brilliant*; *generous.*
And as a person: *modest*; *fun*; *a great
cook.* While I cannot vouch for the veracity of the last attribute,
the first four are certainly to the point.

Dear Luis, actually what I have been trying to say is that I admire your work immensely. On behalf of the Royal Swedish Academy of Sciences I extend to you our warm congratulations. Please step forward to receive the 2005 Rolf Schock Prize in Mathematics from the hands of Princess Christina.

*Christer Kiselman*