## International Workshop on Elliptic and Kinetic Partial Differential Equations

Mini Courses Week 1:

Prof. Lihe Wang (Iowa)

Regularity theory of elliptic equations - Abstract: The first lecture will start out with the maximum principle and their gradient estimates. Most importantly we will introduce the idea of geometrical interpretation of the estimate and its scaling.We will talk about the weak formulation of the elliptic equation. Second lecture is about Schauder estimate. Both maximum principle approach and the regularity of weak solution will be represented. We will present the DeGiorgi and Nash theory in the last lecture. We will discuss the geometrical interpretation along each of the computations.

Prof. Luigi Ambrosio (Scuola Normale Superiore - Pisa)

Flow of nonsmooth vector fields and applications. Part I - Abstract: At the beginning of the 1990→s, DiPerna and Lions made a deep study on the connection between transport equations and ordinary differential equations. In particular, by proving existence and uniqueness of bounded solutions for transport equations with Sobolev vector-fields, they obtained (roughly speaking) existence and uniqueness of solutions for ODEs for a.e. initial condition. Ten years later, Ambrosio extended this result to BV vector fields, providing also a new axiomatization of the theory of flows, more based on probabilistic tools. In recent years, several new extensions have been obtained, that give rise to applications to PDEs which include some systems of conservation laws, semi-geostrophic equations, the linear Schrodinger equation and the Vlasov-Poisson equation. In the first part of the lectures (by L. Ambrosio), we will introduce the general theory of flows, covering the duality between the ODE well-posedness and the PDE well-posedness and presenting basic classes of vector fields (Sobolev, BV, ...) where this theory applies. In the second part of the lectures (by A. Figalli), we shall focus on the more recent extensions and their applications to PDEs.

Prof. Laure Saint-Raymond (École Normale Supérieure - Paris)

From particle systems to collisional kinetic equations - Abstract: 1. The low density limit: formal derivation Consider a deterministic system of $N$ hard spheres of diameter $\eps$. Assume that they are initially independent and identically distributed. Then, in the limit when $N\to \infty$ and $\eps \to 0$ with $N\eps^2=1$ (Boltzmann-Grad scaling), the one-particle density can be approximated by the solution to the kinetic Boltzmann equation. In particular, particles remain asymptotically independent. In the first lecture, we will present the formal derivation of this low density limit, and discuss two important features, namely the propagation of chaos and the appearance of irreversibility. 2. A short time convergence result Lanford's theorem states that in the Boltzmann-Grad limit the one-particle density converges to the solution of the kinetic Boltzmann equation almost everywhere on a short time interval (corresponding actually to a fraction of the average first collision time). The proof relies on a careful study of the recollision mechanism (which is not described by the Boltzmann dynamics), and on a priori bounds obtained by a Cauchy-Kowalewski argument. In the second lecture, we will give a sketch of this proof, and show that the time restriction is due to the lack of global a priori bounds

Prof. Arshak Petrosyan (Purdue)

The Thin Obstacle Problem - Abstract: We will discuss the techniques in the study of the thin obstacle problem, developed in the recent years: - Almgren's, Weiss's, and Monneau's monotonicity formulas; - Epiperimetric inequality; - Partial hodograph-Legendre transform (and connection with subelliptic equations) - Higher order boundary Harnack principle of De Silva - Savin.

#### Contributed Talks

Friday - July 10th

Nome: Dennis Kriventsov (University of Texas at Austin - USA)
Título: Free boundary problem related to thermal insulation

Nome: Robin Neumayer (University of Texas at Austin - USA)
Título: A strong form of the quantitative Wulff inequality

Nome: Juliana F.S. Pimentel (ICMC - USP - Brazil)
Título: Asymptotic behavior of nondissipative scalar reaction-diffusion equations

Nome: Javier Morales (University of Texas at Austin - USA)
Título: Self propelled particles, Optimal transportation and the logarithmic Sobolev inequality

Nome: Roberto Velho (KAUST - Saudi Arabia)
Título: A Short Introduction to Mean Field Games

Nome: David Evangelista da Silveira Junior (KAUST - Saudi Arabia)
Título: Generalised mean-field games with congestion

Nome: Léonard Monsaingeon (University of Texas at Austin - IST-Lisbon)
Título: A new optimal transport distance between nonnegative measures in R^d

Nome: Maja Taskovic (University of Texas at Austin - USA)
Título: Exponential tails for solutions to the homogeneous Boltzmann equation

Nome: Edgard Pimentel (UFC & UFSCAR - Brazil)
Título: Elliptic Mean Field Games Systems

Nome: Juan Spedaletti (Universidad de San Luis - CONICET Argentina)
Título: Convergence Results for the Steklov eigenvalue and optimal windows in Oscillating Domains

Nome: Rohit Jain (University of Texas at Austin - USA)
Título: The Fully Nonlinear Stochastic Impulse Control Problem

Mini Courses Week 2:

Prof. Henri Berestycki (CNRS/EHESS - Paris)

Reaction-diffusion and propagation in non-homogenous media - Abstract: The classical theory of reaction-diffusion deals with nonlinear parabolic equations that are homogenous in space and in time. It analyses travelling waves, long time behavior and the speed of propagation. More general, heterogeneous reaction-diffusion equations arise naturally in models of ecology, biology and medicine that lead to challenging mathematical questions. In this series of lectures, after reviewing fundamental results of the classical theory, I will describe recent progress on models that involve spatially heterogeneous non-linear parabolic and elliptic equations. I will also consider cases with non-local diffusions. The course will involve the following themes: 1. Review of the classical theory of homogenous reaction-diffusion equations. 2. The effect of a line with fast diffusion on Fisher-KPP invasion. 3. The effect of domain shape. 4. Models with non-local operators. 5. Propagation and spreading speeds in non-homogeneous media.

Prof. Irene Gamba (Texas - Austin)

Analytical issues for non-local multi-linear interaction models: The Boltzmann and related equations - Abstract: The Boltzmann equation models the evolution of continuum random processes for multilinear dynamics. Nowadays, diverse models based on the ideas of Boltzmann and Maxwell, referred also as collisional kinetic transport in particle interacting systems, are widely used in modeling phenomena ranging from rarefied classical gas dynamics, inelastic interacting systems in granular or polymer kinetic flows, collisional plasmas and electron transport in nanostructures in mean field theories, to self-organized or social interacting dynamics. This type of models share a common description based in a Markovian framework of birth and death processes in a multi-linear setting. Following the Introductory lectures “From particle systems to collisional kinetic equations” by Laure Saint-Raymond, in the first two lectures, we will focus on diverse analytical issues depending on the properties of the transition probability rates associated to the Markovian process. We will discuss both the space homogeneous as well as the inhomogeneous problems. The results strongly depend on the structure of transition probability rates, which controls regularity, high-energy tail properties, as well as long time behavior of the solutions to steady or self-similar states. We will also discuss the Coulomb potential limit case the yields the Landau equation widely use in collisional plasma theory. The third lecture will focus on an interesting result that distinguish the characterization of space inhomogeneous stationary solutions of in all space vs. a tori, where the effects of dispersion and dissipation interplay producing unexpected effects. We analyze the existence and long time behavior of solutions of the space inhomogeneous Boltzmann equation in the whole space for initial data in the vicinity of a global Maxwellian, and show, surprisingly, the existence of a scattering regime that leads to the construction of eternal solutions that do not converge to such global Maxwellian, as expected from the H-theorem.

Prof. Alessio Figalli (Texas - Austin)

Flow of nonsmooth vector fields and applications. Part II - Abstract: At the beginning of the 1990s, DiPerna and Lions made a deep study on the connection between transport equations and ordinary differential equations. In particular, by proving existence and uniqueness of bounded solutions for transport equations with Sobolev vector-fields, they obtained (roughly speaking) existence and uniqueness of solutions for ODEs for a.e. initial condition. Ten years later, Ambrosio extended this result to BV vector fields, providing also a new axiomatization of the theory of flows, more based on probabilistic tools. In recent years, several new extensions have been obtained, that give rise to applications to PDEs which include some systems of conservation laws, semi-geostrophic equations, the linear Schrodinger equation and the Vlasov-Poisson equation. In the first part of the lectures (by L. Ambrosio), we will introduce the general theory of flows, covering the duality between the ODE well-posedness and the PDE well-posedness and presenting basic classes of vector fields (Sobolev, BV, ...) where this theory applies. In the second part of the lectures (by A. Figalli), we shall focus on the more recent extensions and their applications to PDEs.