# Variational problems of isoperimetric type. Stability and Geometric flows (research costs) (SAiso)

Main funder

Funder's project number: 314227

Funds granted by main funder (€)

- 210 000,00

Funding program

Project timetable

Project start date: 01/09/2017

Project end date: 31/08/2020

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.

Principal Investigator

Primary responsible unit

Related publications

- A non local approximation of the Gaussian perimeter : Gamma convergence and Isoperimetric properties (2021) De Rosa, Antonio; et al.; A1; OA
- A quantitative reverse Faber-Krahn inequality for the first Robin eigenvalue with negative boundary parameter (2021) Cito, Simone; et al.; A1
- Volume preserving mean curvature flows near strictly stable sets in flat torus (2021) Niinikoski, Joonas; A1; OA
- On the regularity of very weak solutions for linear elliptic equations in divergence form (2020) La Manna, Domenico Angelo; et al.; A1; OA
- Short time existence of the classical solution to the fractional mean curvature flow (2020) Julin, Vesa; et al.; A1; OA
- Symmetry of minimizers of a Gaussian isoperimetric problem (2020) Barchiesi, Marco; et al.; A1; OA
- The Surface Diffusion Flow with Elasticity in Three Dimensions (2020) Fusco, Nicola; et al.; A1; OA