# Variational problems of isoperimetric type. Stability and Geometric flows. (research cost) (SAiso2)

Main funder

Funder's project number: 335547

Funds granted by main funder (€)

- 159 999,00

Funding program

Project timetable

Project start date: 01/09/2020

Project end date: 31/08/2022

Summary

The isoperimetric inequality states that balls have the smallest surface area among all sets with same volume. This is a fundamental

result in geometric measure theory, since it provides a link between analysis and geometry. The stability of the isoperimetric inequality

means to study the following question: if the surface area of a set is close to the surface area of the ball, does the set look like the ball?

The goal of this project is to establish new stability results for relevant isoperimetric inequalities.

These results give new information on minimizers of geometric variational problems which are characterized by a competition of

short-range attractive and long-range repulsive forces. Such problems arise for instance in material physics and in nuclear physics. We

will also study the stability of geometric flows associated with these energies. In particular, the aim is to study the behavior of these

flows near local minimum points. The stability of geometric inequalities is also used to obtain sharp quantitative estimates for

homogenization of Faber-Krahn type problems.

result in geometric measure theory, since it provides a link between analysis and geometry. The stability of the isoperimetric inequality

means to study the following question: if the surface area of a set is close to the surface area of the ball, does the set look like the ball?

The goal of this project is to establish new stability results for relevant isoperimetric inequalities.

These results give new information on minimizers of geometric variational problems which are characterized by a competition of

short-range attractive and long-range repulsive forces. Such problems arise for instance in material physics and in nuclear physics. We

will also study the stability of geometric flows associated with these energies. In particular, the aim is to study the behavior of these

flows near local minimum points. The stability of geometric inequalities is also used to obtain sharp quantitative estimates for

homogenization of Faber-Krahn type problems.