Courses (4 hours each) by
Universidad Autónoma de Madrid
Title: Nonlinear and Nonlocal Degenerate Diffusions on Bounded Domains
Description: Nonlinear diffusion models appear in several real world phenomena, ranging from physics, engineering and information theory to life sciences and finance. This mini course will be focussed on a series of recent papers in collaboration with A. Figalli, X. Ros Oton, Y. Sire and J. L. Vázquez. We develop a complete theory for a diffusion model of Porous Medium type, with different nonlocal operators and degenerate nonlinearities. The first lectures will be devoted to the basic theory: duality approach by means of Green functions, definition of weak dual solutions, existence, uniqueness and smoothing effects. In the second part of the course we will address more advanced topics: depending on time and interest of the audience, we will focus on some topics among: Harnack inequalities, Hölder continuity, higher regularity estimates, sharp boundary regularity, sharp asymptotic behaviour. We will pay attention to a surprising anomalous boundary behaviour that appears because of both the degeneracy of the nonlinearity and the nonlocal character of the equation.
Università di Milano Bicocca
Title: Monotonicity methods and unique continuation for fractional elliptic equations.
Description: In the first part of this mini-course, monotonicity methods for the study of unique continuation principles for elliptic and parabolic operators will be introduced.
In the second part, these methods will be applied to fractional elliptic equations, for which it will be shown how a combination of monotonicity formulas with blow-up analysis allows obtaining a precise description of possible blow-up profiles in terms of a Neumann eigenvalue problem on the sphere.
Norwegian University of Science and Technology
Title: Numerical approximations of nonlocal equations
Description: For scalar conservation laws there is a well-established philosophy to prove prove convergence monotone numerical schemes. The key ingredients are uniform $L^\infty$ and spatial $L^1$ translation estimates, equi-continuity type estimates in time, compactness theorems for $L^1$ and $C_b$, and weak stability result for weak solutions of the conservation law. The resulting convergence will be in $L^1$ or $L^1_loc$ in space.
Such a strategy requires minimal regularity of solutions of the original problem (here they can form shock-discontinuities), and even if more regularity is available, it still gives a very efficient and minimalistic way to obtain strong convergence.
In this course we will explain how this philosophy can be adapted to equation with fractional and nonlocal diffusion. We will focus on nonlinear equations of porous medium type with minimal regularity assumptions on the nonlinearities and data. If time permits, fractional conservation laws can also be considered. The methods we introduce will be applicable to slow and fast diffusions and strongly degenerate Stefan type problems. We will also explain how to construct several approximations that falls into this framework. Several examples will be given.
University of Pittsburgh
Title: A short course on Tug of War games
Description: In five lectures, we will present an overview of the basic constructions pertaining to the recently emerged field of Tug of War games with noise, as seen from an analyst’s (rather than probabilist's) perspective.
University of Wrocław
Title: Concentration phenomena and non-local evolution equations
Description: In this series of lectures, recent results on properties of solutions to nonlinear evolution equations involving either the fractional Laplacian or other non- local “diffusion”-type operators will be presented. The following topics will be discusses:
• blow-up of solutions to non-local models of chemotaxis;
• singular solutions of drift-diffusion equations;
• stable discontinuous stationary solutions of non-local models of pattern formation.