Short courses

Title: The geometry of level sets 

Julie Clutterbuck, Monash University

Abstract: The level set of a function $u$ is the curve or surface $u^{-1}(c)$.  If a positive function is a solution to an elliptic equation on a bounded domain, the shape of its level sets are necessarily constrained.  Investigating the shapes of the level sets has been a fruitful strand in PDE over the last few decades, leading to tools that are used in many applications.  The archetypical result here is that the superlevel sets of the first eigenfunction of the Dirichlet Laplacian are convex.  We’ll look at a few different proof techniques (maximum principles, constant-rank theorems) as well as some open problems in the field.

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Title: Introduction to Mathematical Control Problems

Luz de Teresa, Federal University of Paraíba and Universidad Nacional Autónoma de México

Abstract: In this minicourse, I will introduce some control problems that are treated from the mathematical point of view. We will present some results and techniques related with Ordinary and  Partial Differential Equations.

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Title: Regularity results for elliptic equations involving regional fractional Laplacian

Mouhamed Fall, African Institute for Mathematical Sciences

Abstract: We consider  nonlocal operators generated by symmetric stable processes describing motions of random particles in a region Omega which are censored to jump outside Omega. While particles can jump inside the domain, they  are either reflected inside   or killed when they reach the boundary of the domain. Atypical examples of  such operators are the  “regional fractional laplacians”. In the Brownian case these phenomenon can be described by the Laplace operator subject to Neumann or Dirichlet boundary conditions. A natural generalization to symmetric stable Lévy processes  was found by Bogdan, Burdzy and Chen. In this lecture, we shall investigate the properties of  these nonlocal operators and the regularity results of solutions to the corresponding Poisson problem.


Title: Spontaneously stochastic solutions

Alexei A. Mailybaev, IMPA

Abstract: Models of Fluid Dynamics that lack uniqueness of solutions arise in studies of turbulence. We consider such models within a broader context of differential equations that lack Lipschitz continuity and, therefore, require a regularization procedure or a selection criterion for defining relevant solutions. Starting with simplified models, we show how spontaneously stochastic solutions arise in formally deterministic systems. This phenomenon occurs when one takes into account an infinitesimal noise in the regularization process. Then we present theoretical and numerical results demonstrating spontaneous stochasticity in realistic models of fluid dynamics.


Title: Nonlocal Hénon equation in R^N

Alexander Quaas, Universidad Técnica Federico Santa María



Title: A broad look at elliptic partial differential equations

Enrico Valdinoci, University of Western Australia

Abstract: We will discuss some classical (and less classical) topics related to elliptic partial differential equations, related also to symmetry, classification and regularity theory. Some applications arising in geometry and material sciences will be also presented.

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