Advanced course on Geometric Analysis

Date: September 14 – 18,  2015.
Place:  CRM  (Centre de Recerca Matemàtica, Barcelona)
Inscription:  FREE but compulsory.

Timetable
09.30 to 11.00:
11.00 to 11.45:
11.45 to 13.00:
13.00 to 15.00:
15.00 to 16.15:		 Monday, 14:
Koskela
Coffee break
Csörnyei
Lunch
Mingione		 Tuesday, 15:
Koskela
Coffee break
Csörnyei
Lunch
Mingione		 Wednesday, 16:
Koskela
Coffee break
Csörnyei
Lunch
Mingione		 Thursday, 17:
Koskela
Coffee break
Minigione
Lunch
Csörnyei		 Friday, 18:
Mingione
Coffee break
Csörnyei
.
.

– Marianna Cörnyei (Univ. Chicago)
“Tangents of sets and differentiability of functions”

Abstract:
One of the classical theorems of Lebesgue tells us that Lipschitz functions on the real line are differentiable almost everywhere. We study possible generalisations of this theorem and some interesting geometric corollaries.

– Pekka Koskela (Univ. Jyväskylä)
“Sobolev spaces on simply connected planar domains”

Abstract:
Smooth functions are dense in a first order Sobolev space of a domain, but it need not be the case that restrictions of entire smooth  functions are. This turns out to be case when the domain in question is a planar Jordan domain. In the less restrictive case of a bounded simply connected planar domain, the density of restrictions of entire smooth functions may fail, but still bounded smooth functions with bounded first order derivatives are dense. I will explain the reasons behind these results. The density of entire smooth functions follows trivially if each function in our Sobolev space admits an extension to an entire Sobolev function. I will give geometric characterizations for (bounded) simply connected planar domains which have this property.

– Giuseppe Mingione  (Univ. Parma)
“Recent progresses in nonlinear potential theory”   Slides of the course

Abstract:
The classical potential theory deals with fine and regularity properties of harmonic functions and more in general of solutions to linear elliptic and parabolic equations. In particular, pointwise behaviour of solutions in terms of the data and size estimates of singular sets are at the center of the analysis. Nonlinear potential theory is essentially concerned with the same problems, but when one is considering nonlinear equations. Over the last years there has been a great deal of activities in this direction and I would like to give a survey of some recent results from the subject.