2025

The Analysis School 2025 will take place from June 2nd to June 6th, 2025, with a break scheduled on Wednesday. The event will offer both on-site and online participation via Zoom. All lectures will be recorded and made available for later viewing, subject to the lecturers' consent.

Nicola Fusco (University of Naples Federico II, Italy)

Title: "Isoperimetric type inequalities for capillary surfaces“

Abstract: The study of capillary energies is a classical topic in Calculus of Variation. The course will start with a quick revue of basic definitions and properties of sets of finite perimeter. This will provide us with the tools needed to show the existence of minimizers for the capillary energy and to characterize the solution of the classical problem of the sessile drop sitting on a half space. Then we will present a new shorter proof of a well-known isoperimetric type result of Choe-Ghomi and Ritor\'e, stating that half balls are the unique minimizers of the relative isoperimetric inequality outside a convex set. We will also discuss the same problem in the case of capillary surfaces with a general contact angle lying outside convex infinite cylinders. In particular we will see that the capillary energy of any surface supported on any such convex cylinder is strictly larger than that of a spherical cap with the same volume and the same contact angle on a flat support, unless the surface is itself a spherical cap resting on a facet of the cylinder. We will conclude the course by presenting some recent results on the behaviour of the isoperimetric profile in the exterior of a convex body for large volumes.

Joaquim Serra (ETH Zürich, Switzerland)

Title: "Nonlocal minimal surfaces“

Abstract: We will give an overview of nonlocal minimal surfaces in Euclidean space and on Riemannian manifolds.

Tentative outline:

1) The Caffarelli-Roquejoffre-Savin regularity theory for minimizers.

2) The a priori curvature estimates for nonlocal minimal surfaces of finite index.

3) The robust regularity of stable s-minimal surfaces as s approaches 1.

4) Applications to minmax.

5) What are higher codimension nonlocal minimal surfaces?

Organizers:

Yi Zhang (Chinese Academy of Sciences), Yuan Zhou (Beijing Normal University)

Schedule and Location:

Location and Dates: N913, South Building of the AMSS (June 2–6, with a break on Wednesday).

Zoom meeting: 872 0683 6498 Code: AMSS2025

On-site Course (Fusco): 9:00–11:00 AM (Beijing Time).

Online Course (Serra): 3:00–5:00 PM (Beijing Time; 9:00–11:00 AM Zurich Time).

Both on-site and online streaming options will be available for all lectures.