# Mathematical Sciences Research Institute

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# Workshop

Introductory Workshop in Nonlinear Elliptic Equations and Its Applications August 15, 2005 - August 19, 2005
 Registration Deadline: August 01, 2005 over 8 years ago May 15, 2005 almost 9 years ago
Organizers Luis Caffarelli, L. Craig Evans, Matt Gursky, Cristian Gutierrez, Paul Yang
Speaker(s)

## Show List of Speakers

Description

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”

Funding & Logistics

## Show Funding

To apply for funding, you must register by the funding application deadline displayed above.

Students, recent Ph.D.'s, women, and members of underrepresented minorities are particularly encouraged to apply. Funding awards are typically made 6 weeks before the workshop begins. Requests received after the funding deadline are considered only if additional funds become available.

## Show Lodging

MSRI has preferred rates at the Rose Garden Inn, depending on room availability. Reservations may be made by calling 1-800-992-9005 OR directly on their website. Click on Corporate at the bottom of the screen and when prompted enter code MATH (this code is not case sensitive). By using this code a new calendar will appear and will show the MSRI rate on all room types available.

MSRI has preferred rates at the Hotel Durant. Reservations may be made by calling 1-800-238-7268. When making reservations, guests must request the MSRI preferred rate. If you are making your reservations on line, please go to this link and enter the promo/corporate code MSRI123. Our preferred rate is $129 per night for a Deluxe Queen/King, based on availability. MSRI has preferred rates of$149 - $189 plus tax at the Hotel Shattuck Plaza, depending on room availability. Guests can either call the hotel's main line at 510-845-7300 and ask for the MSRI- Mathematical Science Research Inst. discount; or go to www.hotelshattuckplaza.com and click Book Now. Once on the reservation page, click “Promo/Corporate Code“ and input the code: msri. MSRI has preferred rates of$110 - \$140 at the Berkeley Lab Guest House, depending on room availability. Reservations may be made by calling 510-495-8000 or directly on their website. Select “I am an individual traveler affiliated with MSRI”.

Additional lodging options may be found on our short term housing page.

## Show Visa/Immigration

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