## Overview

This intensive series of short courses will bridge the theoretical foundations of nonlinear partial differential equations (PDEs) with real-world applications. Renowned speakers include Lawrence Craig Evans (UC Berkeley), Italo Capuzzo-Dolcetta (Rome), and Jose Carrillo (Oxford). Topics covered will include viscosity solutions, Hamilton-Jacobi equations, mean field games, kinetic theory, models of collective behavior, and regularity theory for elliptic and parabolic PDEs.

## Course Schedule

Date | 3/11 (Sunday) | 4/11 (Monday) | 5/11 (Tuesday) |
---|---|---|---|

9:00 - 10:30 |
Prof. Carrillo | Prof. Capuzzo-Dolcetta | Prof. Capuzzo-Dolcetta |

10:30 - 12:00 |
Prof. Evans | Prof. Evans | Prof. Carrillo |

## Speaker and Talk Details

### Global Minimizers of Interaction Energies

#### Speaker

Dr. Jose A. Carrillo, Professor of the Analysis of Nonlinear Partial Differential Equations, Mathematical Institute, University of Oxford.

#### Abstract

I will review the existence and uniqueness of global minimizers for interaction energy functionals. Euler-Lagrange equations in the infinity wasserstein distance will be discussed. Based on linear convexity/concavity arguments, qualitative properties of the global minimizers will also be treated. Anisotropic singular potentials appearing in dislocations will be shown to have rich qualitative properties with loss of dimension and ranges of explicit minimizers. This talks will be based on several works in collaboration with Ruiwen Shu (University of Oxford).

#### Biography

José Antonio Carrillo is a distinguished mathematician who has made fundamental contributions to partial differential equations (PDEs) and their applications across physics, biology, and social sciences. His research spans kinetic theory, nonlinear aggregation-diffusion equations, and collective behavior models.

After completing dual bachelor's degrees in Mathematics and Computer Science at the University of Granada in 1992, Carrillo earned his Ph.D. in Mathematics there in 1996. He held positions at the University of Texas at Austin and the University of Granada before becoming an ICREA Research Professor at the Universitat Autònoma de Barcelona in 2003. In 2012, he joined Imperial College London as Chair in Applied and Numerical Analysis, and in 2020 moved to his current position at the University of Oxford, where he also serves as Tutorial Fellow in Applied Mathematics at The Queen's College.

Carrillo's work is characterized by its breadth, connecting pure mathematical analysis with real-world applications. He has made pioneering contributions to the mathematical modeling of collective behavior, developing frameworks to understand phenomena from cell organization to crowd dynamics. His research on nonlocal PDEs and gradient flows has opened new avenues in the field.

His excellence in research has been recognized through numerous honors, including election to the European Academy of Sciences (2018), the Richard von Mises Prize (2006), and being named a SIAM Fellow (2019). He currently holds an ERC Advanced Grant for his work on nonlocal PDEs for complex particle dynamics. As Chair of the Applied Mathematics Committee of the European Mathematical Society (2014-2017) and through his extensive editorial work, including roles at SIAM Journal on Mathematical Analysis and Proceedings of the London Mathematical Society, he has helped shape the direction of applied mathematics research.

Related website: https://www.maths.ox.ac.uk/people/jose.carrillodelaplata

### Streamlines and Shocks for the Infinity Laplacian PDE

#### Speaker

Lawrence Craig Evans, Professor of Mathematics, University of California, Berkeley.

#### Abstract

I will first discuss several ways by which we can transform nonlinear partial differential equations into linear equations. I will then show how these ideas let us study the geometry of streamlines and shocks for solutions of the two-dimensional infinity Laplacian equation, a very strange PDE that arrises from sup-norm variational problems.

#### Biography

Lawrence Craig Evans is a preeminent mathematician who has made transformative contributions to nonlinear partial differential equations, particularly in the theory of elliptic equations. His foundational work with Nikolai V. Krylov on concave, fully nonlinear, uniformly elliptic equations - for which they shared the 2004 Leroy P. Steele Prize - fundamentally advanced the field by proving solutions are C2,α. Evans has also made seminal contributions to viscosity solutions of nonlinear equations, stochastic optimal control through the Hamilton-Jacobi-Bellman equation, and the theory of harmonic maps.

Born in Atlanta in 1949, Evans completed his undergraduate studies at Vanderbilt University in 1971 before earning his Ph.D. from UCLA in 1975 under Michael G. Crandall. After positions at the University of Kentucky (1975-1980) and the University of Maryland (1980-1989), he joined UC Berkeley in 1989, where he continues to serve as Professor of Mathematics.

His influential graduate textbook Partial Differential Equations has become the standard introduction to the field. He also co-authored Measure Theory and Fine Properties of Functions with Ronald Gariepy, an authoritative work on Hausdorff measure, capacity, Sobolev functions, and sets of finite perimeter.

Evans' exceptional contributions have been recognized through numerous honors, including election to the American Academy of Arts and Sciences (2003), the National Academy of Sciences (2014), and designation as an AMS Fellow (2013). Most recently, he was awarded the 2023 Steele Prize for Mathematical Exposition.

Related website: https://math.berkeley.edu/~evans/

### Compactness and the Curvature of 3-Webs

#### Speaker

Dr. Lawrence Craig Evans, Professor of Mathematics, University of California, Berkeley.

#### Abstract

I will explain some so-called ``trilinear compensated compactness'' methods and outline a new proof of an important theorem of Joly, Metivier and Rauch. The problem is to understand the weak limit of the product of 3 functions, each of which itself converges weakly. These functions solve simple linear ODE, and their trajectories form a 3-web in the plane, with non vanishing curvature.

#### Biography

Lawrence Craig Evans is a preeminent mathematician who has made transformative contributions to nonlinear partial differential equations, particularly in the theory of elliptic equations. His foundational work with Nikolai V. Krylov on concave, fully nonlinear, uniformly elliptic equations - for which they shared the 2004 Leroy P. Steele Prize - fundamentally advanced the field by proving solutions are C2,α. Evans has also made seminal contributions to viscosity solutions of nonlinear equations, stochastic optimal control through the Hamilton-Jacobi-Bellman equation, and the theory of harmonic maps.

Born in Atlanta in 1949, Evans completed his undergraduate studies at Vanderbilt University in 1971 before earning his Ph.D. from UCLA in 1975 under Michael G. Crandall. After positions at the University of Kentucky (1975-1980) and the University of Maryland (1980-1989), he joined UC Berkeley in 1989, where he continues to serve as Professor of Mathematics.

His influential graduate textbook Partial Differential Equations has become the standard introduction to the field. He also co-authored Measure Theory and Fine Properties of Functions with Ronald Gariepy, an authoritative work on Hausdorff measure, capacity, Sobolev functions, and sets of finite perimeter.

Evans' exceptional contributions have been recognized through numerous honors, including election to the American Academy of Arts and Sciences (2003), the National Academy of Sciences (2014), and designation as an AMS Fellow (2013). Most recently, he was awarded the 2023 Steele Prize for Mathematical Exposition.

Related website: https://math.berkeley.edu/~evans/

### Around the Maximum Principle

#### Speaker

Dr. Italo Capuzzo Dolcetta, Full Professor of Mathematics, Facoltà di Scienze, Università di Roma La Sapienza.

#### Abstract

The lectures review the validity, at various degrees of generality, of the Maximum Principle (MP in short) for elliptic operators.

A tentative list of the topics that will be touched is as follows:

- Some elementary cases: linear, convex and subharmonic functions.
- MP for linear uniformly elliptic operators in bounded domains. Alexandrov-Bakelman-Pucci (ABP) Theorem.
- Generalized subharmonic functions, supersolutions in the viscosity sense. A fully nonlinear version of the ABP Theorem.
- MP for fully nonlinear operators on some unbounded domains.
- Validity of MP and positivity of the principal eigenvalue: results in some nonlinear cases.
- MP for weakly coupled linear elliptic systems. Invariant cones for some non weakly coupled systems.
- A short digression in linear algebra and numerics: Perron-Frobenius and Collatz-Wielandt type representation formulas.

#### Biography

Italo Capuzzo Dolcetta has been a Full Professor in Mathematics at the Facoltà di Scienze, Università di Roma La Sapienza, Dipartimento di Matematica since 1987.

##### Scientific Activities

CD's research activity took place mainly n the field of viscosity and variational methods in nonlinear partial differential equations and their applications to control theory and computer vision. He is the co-author with M.Bardi of the book Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations and of more than 90 scientific papers on international journals. He collaborated with several italian and foreign mathematicians among which H. Berestycki, L. C. Evans, H. Ishii, P.L.Lions, L. Nirenberg. B. Perthame.

##### Research Visits

CD has been visiting professor in several international research institutions such as:

Department of Mathematics (University of Maryland), Laboratoire d'Analyse Numerique (Université Paris VI, CEREMADE (Université Paris IX), INRIA (1981), Institut fur Mathematik (Università di Bonn), Banach Center (Accademia Polacca delle Scienze), Department of Mathematics (Chuo University, Tokyo), Division of Applied Mathematics (Brown University, Department of Mathematics (University of British Columbia), Universitè d’ Orleans, KAUST, Université de Rouen, University of Tubingen, Courant Institute, NYU, Tokyo Metropolitan University, University of Texas at Austin, Universitè de Rennes, Waseda University (Tokyo), EPFL (Lausanne), ETH (Zurich).

##### Management of Research Activities

ICD has been the principal investigator of several Italian and European research programs and Director of the Istituto Nazionale di Alta Matematica section Analysis, Probability and Applications.

##### Editorial Activities

Chairman of Nonlinear Differential Equations and Applications (NoDEA). Member of the Editorial Board of ESAIM - Control and Calculus of Variations Member of the Editorial Board of Rendiconti di Matematica. Member of the Editorial Board of Minimax Theory and its Applications