# The DiPerna Lectures

The DiPerna Lecture was established by friends and colleagues as a memorial to Ronald DiPerna after his untimely death 1989. Each year the Department of Mathematics invites an outstanding applied mathematician to deliver the DiPerna Lecture on recent advances in applied mathematics.

### Title: A Variational Regularity Theory for Optimal Transportation

Abstract:  The optimal transportation of one measure into another, leading to the notion of their Wasserstein distance, is a problem in the calculus of variations with a wide range of applications. The regularity theory for the optimal map is subtle and was revolutionized by Caffarelli. This approach relies on the fact that the Euler-Lagrange equation of this variational problem is given by the Monge-Ampére equation. The latter is a prime example of a fully nonlinear (degenerate) elliptic equation, amenable to comparison principle arguments.

We present a purely variational approach to the regularity theory for optimal transportation, introduced with M. Goldman and refined with M. Huesmann. Following De Giorgi's philosophy for the regularity theory of minimal surfaces, it is based on the approximation of the displacement by a harmonic gradient, through the construction of a variational competitor.  This leads to a one-step improvement lemma'', and feeds into a Campanato iteration on the C1,α-level for the optimal map, capitalizing on affine invariance.

### Ronald J. DiPerna, 1947-1989

The following text is from the memorial written for Professor DiPerna shortly after his death:

Ronald J. DiPerna, a professor of mathematics at the University of California at Berkeley, died in Princeton on January 8, 1989. At the time of his death he was a visiting member of the Institute for Advanced Study at Princeton, NJ. His wife, Maria Schonbeck, is a professor of mathematics at the University of California, Santa Cruz. They have one daughter, Lauren.

DiPerna was born in Sommerville, Massachusetts, on February 11, 1947. He received his Ph.D. at the Courant Institute of Mathematical Sciences of New York University in 1972, and held faculty positions at Brown University, the University of Michigan, the University of Wisconsin and Duke University before coming to Berkeley in 1985.

DiPerna was known for his work on nonlinear partial differential equations, especially those that are important in fluid dynamics and the kinetic theory of gases. Probably his best known work is his development and application of the method of compensated compactness. This is a very powerful method for controlling oscillation and thereby proving existence theorems. DiPerna proved existence of weak solutions in the large for the equations of compressible gas dynamics and obtained important results concerning the uniqueness of solutions, their large time behavior, and their local regularity as elements of the appropriate abstract spaces.

His recent work concerned integro-differential equations that arise in the kinetic theory of gases and certain types of singularity that arise in incompressible flow.

DiPerna's work is remarkable for the courage and vision with which he attacked and conquered problems of exceptional difficulty. His papers are masterpieces of hard analysis and a source of wonder and inspiration to all those who read them and learn from them. His dedication to mathematics is legendary, and his scientific vision permeates much of contemporary analysis. His very premature death deprives the mathematics community as a whole, and the mathematics department at Berkeley in particular, of a very important and innovative voice.

DiPerna held Guggenheim and Sloan fellowships; he spoke at the International Congress of Mathematicians, and was a member of the Institute for Advanced Study at Princeton. Had he lived, his work would have earned the highest honors that the mathematical community can bestow. The Mathematics Department at Berkeley keenly feels his loss.

Alexandre J. Chorin
Craig Evans
James Gilman
Andrew Majda

### Past DiPerna Lecturers

2020 Carlos Kenig
2019 Gunther Uhlmann
2018 Wilhelm Schlag
2017 Herbert Koch
2016 Gilles Lebeau
2014 Takis Souganidis
2013 Alan Newell
2012 Emmanuel Candes
2010 Lai-Sang Young
2009 Andrew J. Majda
2008 Gui-Qiang Chen
2007 Yuxi Zheng
2006 Alberto Bressan
2005 Charles Fefferman
2004 Benoit Perthame
2003 John Ball
2002 Andrew Stuart
2001 Ciprian Foias