Minicourses:

This is a three day long summer school following the workshop on Nonlinear Elliptic PDE.  

Place: Aula Azul, ICMAT

There will be three advanced minicourses (4h30m each) by:

• Veronica Felli (Univ. Milano Bicocca, IT)

• Giuseppe Mingione (Univ. Parma, IT)

• Juncheng Wei (Univ. British Columbia, Vancouver, CA)

The courses will be also posted in the youtube channel of ICMAT

[Click on this link]

Change of time in the afternoon: we begin at 15:30

The registration is free but mandatory for organization issues. To attend, please fill in this form 

Some financial support is available for PhD students and young postdocs (deadline June 10th, 2023). 

Titles and Abstracts of the courses: (and some notes)

• V. Felli: Monotonicity methods and unique continuation for fractional elliptic equations

In the first part of this mini-course, monotonicity methods for the study of unique continuation principles for elliptic 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. 

• G. Mingione: Nonuniform ellipticity 

Nonuniform ellipticity is a classical topic in PDE theory.  It deals with elliptic operators built on positive definite matrixes where, roughly speaking, there is no way to control the largest eigenvalue by the smaller one. The minimal surface operator is one example. More instances come from applied settings connected, for example, with  Homogenization, Nonlinear Elasticity, Non-newtonian Fluid Dynamics. The regularity theory of nonuniformly elliptic operators is also a classical topic, initiated by authors like Bombieri, De Giorgi, Ladyzhenskaya & Uraltseva, Stampacchia, Gilbarg, Trudinger, Serrin, Simon, in the 60/70s. My lectures are aimed at presenting some recent developments in the regularity theory of nonuniformly elliptic operators, with special emphasis on those coming from variational problems. In particular I will present a new approach to Schauder theory, that in the uniformly elliptic setting ceases to be perturbative and, in certain cases, fails to hold. 

• J. Wei: Introduction to parabolic gluing  [Notes of the course]

In this mini-course, I will introduce the various aspects and applications of parabolic gluing inner-outer methods. One key difficulty is the lack of Fredholm properties for parabolic equations. I will start with the L^2-case, which resembles the elliptic equation. Then several methods are introduced to deal with non-L^2 case. Several applications will be discussed, including harmonic map flows and Yang-Mills flow. 

Poster