Mathematical Sciences Research Institute - Introductory Workshop in Nonlinear Elliptic Equations and Its Applications

This workshop will consists of two main series of lectures, one each day by Luis Caffarelli and Matt Gursky. The intention of these series is to give an overview of the central ideas of the program on Nonlinear Elliptic Equations. In addition there will be several individual lectures on specific topics of current research activitiy.

Five lectures by Luis Caffarelli: Basic tools for the regularity theory of nonlinear elliptic equations.
How do nonlinear elliptic PDE fit within the calculus of variations? What are fully nonlinear equations, and what do you need to prove regularity?
Topics will include: Sobolev embedding versus energy inequalities, geometric arguments, the oscillation theorem of De Giorgi, Calderon--Zygmund type decomposition and the Harnak inequality. Some advanced topics will be the Krylov--Safonov Harnack inequality, and perhaps the Evans--Krylov estimates.

Five Lectures by Matt Gursky Some Fully Nonlinear Equations in Geometry
The goal of this series of lectures is to describe some examples of fully nonlinear equations which arise in geometry. Although geometrically inspired problems provided the motivation for much of the development of the modern theory of nonlinear PDEs, we will not attempt an historical survey. Rather, we will concentrate on two model problems to illustrate some of the main ideas and techniques: the problem of isometrically embedding a surface in 3D Euclidean space, and the problem of prescribing the sigma-k curvature of a conformal metric.

Other invited speakers are: Patricia Bauman , Maria Del Mar Gonzalez, Pengfei Guan, Emmanuel Hebey, Nikolai Krylov, Yan Yan Li, Fang-Hua Lin, Robert McCann, Mikhail Safonov, Yu Yuan.

Lecture Schedule

Monday August 15
9:30-10:30 Luis Caffarelli:"Basic tools for the regularity theory of nonlinear elliptic equations, I"
10:30-11:00 Tea Break - 2nd Floor
11:00-12:00 Matt Gursky:"Some Fully Nonlinear Equations in Geometry, I"
12:00-1:30 Lunch
1:30-2:30 Patricia E. Bauman: “Nonlinear PDE Methods for Analysis of Phase Transitions in Superconducting and Liquid Crystals”
2:30-3:30 Andras Domokos: “Regularity of quasilinear subelliptic equations”
3:30-4:00 Tea Break - 2nd Floor
4:00-5:00 Yu Yuan: "Resolving the singularities of minimal Hopf cones"

Tuesday August 16
9:30-10:30 Luis Caffarelli:"Basic tools for the Regularity Theory of Nonlinear Elliptic Equations, II"
10:30-11:00 Tea Break - 2nd Floor
11:00-12:00 Matt Gursky:"Some Fully Nonlinear Equations in Geometry, II"
12:00-1:30 Lunch
1:30-2:30 Mikhail Safonov: “Boundary behavior of solutions to second order elliptic and parabolic equations in non-smooth domains”
2:30-3:30 Diego Maldonado: “On the regularity of solutions to the Monge-Ampere equation and its linearization”
3:30-4:00 Tea Break - 2nd Floor
4:00-5:00 Nikolai Krylov I: "Numerical approximation for fully nonlinear degenerate equations"

Wednesday August 17
9:30-10:30 Luis Caffarelli:"Basic tools for the Regularity Theory of Nonlinear Elliptic Equations, III"
10:30-11:00 Tea Break - 2nd Floor
11:00-12:00 Matt Gursky:"Some Fully Nonlinear Equations in Geometry, III"
12:00-1:30 Lunch
1:30-2:30 Fang-Hua Lin:
2:30-3:30 David A. Hartenstine:
3:30-4:00 Tea Break - 2nd Floor
4:00-5:00 Nikolai Krylov II: "Numerical approximation for fully nonlinear degenerate equations"

Thursday August 18
9:30-10:30 Luis Caffarelli:"Basic tools for the Regularity Theory of Nonlinear Elliptic Equations, IV"
10:30-11:00 Tea Break - 2nd Floor
11:00-12:00 Matt Gursky:"Some Fully Nonlinear Equations in Geometry, IV"
12:00-1:30 Lunch
1:30-2:30 Maria Del Mar Gonzalez:: “Construction of singular solutions for a fully non-linear equation in conformal geometry”
2:30-3:30 Emmanuel Hebey: "Fourth order PDEs of critical growth"
3:30-4:00 Tea Break - 2nd Floor
4:00-5:00 Nikolai Krylov III: "Numerical approximation for fully nonlinear degenerate equations"

Friday August 19
9:30-10:30 Luis Caffarelli:"Basic tools for the Regularity Theory of Nonlinear Elliptic Equations, V"
10:30-11:00 Tea Break - 2nd Floor
11:00-12:00 Matt Gursky:"Some Fully Nonlinear Equations in Geometry, V"
12:00-1:30 Lunch
1:30-2:30 Yanyan Li: ``Some Liouville type theorems''
2:30-3:30 Peng-Fei Guan: “A constant rank theorem of convex solutions of fully nonlinear equations”
3:30-4:00 Tea Break - 2nd Floor
4:00-5:00 Robert J. McCann: “Second-order asymptotics for fast nonlinear diffusions”

Currently Available Videos

Luis Caffarelli , Basic Tools for the Regularity Theory of Nonlinear Elliptic Equations, I August 15,2005, 09:30 AM to 10:30 AM
Matthew Gursky , Some Fully Nonlinear Equations in Geometry, I August 15,2005, 11:00 AM to 12:00 PM
Patricia Bauman , Nonlinear PDE Methods for Analysis of Phase Transitions in Superconducting and Liquid Crystals August 15,2005, 01:30 PM to 02:30 PM
Andras Domokos , Regularity of Quasilinear Subelliptic Equations August 15,2005, 02:30 PM to 03:30 PM
Yu Yuan , Resolving the Singularities of Minimal Hopf Cones August 15,2005, 04:00 PM to 05:00 PM
Luis Caffarelli , Basic Tools for the Regularity Theory of Nonlinear Elliptic Equations, II August 16,2005, 09:30 AM to 10:30 AM
Matthew Gursky , Some Fully Nonlinear Equations in Geometry, II August 16,2005, 11:00 AM to 12:00 PM
Mikhail Safonov , Boundary Behavior of Solutions to Second Order Elliptic and Parabolic Equations in Non-smooth Domains August 16,2005, 01:30 PM to 02:30 PM
Diego Maldonado , On the Regularity of Solutions to the Monge-Ampere Equation and its Linearization August 16,2005, 02:30 PM to 03:30 PM
Nikolai Krylov , Numerical Approximation for Fully Nonlinear Degenerate Equations August 16,2005, 04:00 PM to 05:00 PM
Luis Caffarelli , Basic Tools for the Regularity Theory of Nonlinear Elliptic Equations, III August 17,2005, 09:30 AM to 10:30 AM
Matthew Gursky , Some Fully Nonlinear Equations in Geometry, III August 17,2005, 11:00 AM to 12:00 PM
Fang-Hua Lin , Nonliner Elliptical Equation and Its Application Lecture August 17,2005, 01:30 PM to 02:30 PM
David Hartenstine , A General Perron Method and Application August 17,2005, 02:30 PM to 03:30 PM
Nikolai Krylov , Numerical Approximation for Fully Nonlinear Degenerate Equations-II August 17,2005, 04:00 PM to 05:00 PM
Luis Caffarelli , Basic Tools for the Regularity Theory of Nonlinear Elliptic Equations, IV August 18,2005, 09:30 AM to 10:30 AM
Matthew Gursky , Some Fully Nonlinear Equations in Geometry, IV August 18,2005, 11:00 AM to 12:00 PM
Maria Gonzalez , Construction of Singular Solutions for a Fully Non-linear Equation in Conformal Geometry August 18,2005, 01:30 PM to 02:30 PM
Emmanuel Hebey , Fourth Order PDEs of Critical Growth August 18,2005, 02:30 PM to 03:30 PM
Nikolai Krylov , Numerical Approximation for Fully Nonlinear Degenerate Equations, III August 18,2005, 04:00 PM to 05:00 PM
Luis Caffarelli , Basic Tools for the Regularity Theory of Nonlinear Elliptic Equations, V August 19,2005, 09:30 AM to 10:30 AM
Matthew Gursky , Some Fully Nonlinear Equations in Geometry, V August 19,2005, 11:00 AM to 12:00 PM
Yanyan Li , Some Liouville Type Theorems August 19,2005, 01:30 PM to 02:30 PM
Peng-Fei Guan , A Constant Rank Theorem of Convex Solutions of Fully Nonlinear Equations August 19,2005, 02:30 PM to 03:30 PM
Robert McCann , Second-order Asymptotics for Fast Nonlinear Diffusions August 19,2005, 04:00 PM to 05:00 PM

You can find videos of other workshops and events on our VMath - Streaming Video page.

For more information:
Questions about this workshop should be sent either by email to

299@msri.org

or by regular mail to:

Introductory Workshop in Nonlinear Elliptic Equations and Its Applications Mathematical Sciences Research Institute 17 Gauss Way, Berkeley, CA 94720-5070. USA

The Institute is committed to the principles of Equal Opportunity and Affirmative Action.

Back to Workshop Listing

Want to be kept updated on upcoming events? Then Click Here to Subscribe to Our Newsletters!