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# An introduction to the Cheeger problem III

## Gian Paolo Leonardi Università di Modena

Abstract: the Cheeger problem consists in minimizing the ratio between perimeter and volume among subsets of a given set $\Omega$. The infimum of this ratio is the Cheeger constant of $\Omega$, while minimizers are called Cheeger sets. Quite surprisingly, this variational problem turns out to be closely linked to a number of other relevant problems (eigenvalue estimates, capillarity, image segmentation techniques, max-flow/min-cut duality, landslide models). After introducing some essential concepts and tools from the theory of BV functions and finite perimeter sets, we shall review some classical as well as recent results on this topic.
All lectures will be delivered at the Seminar Room in the 1st floor of the Mathematics building.
Lecture 1. March 20, 12'00–13'30. Introduction. Essentials on BV functions and finite perimeter sets.
Lecture 2: March 21, 16'00–17'30. General properties of Cheeger sets. The two-dimensional case.
Lecture 3: March 22, 12'00–13'30 Links with prescribed mean curvature equation and capillarity.

# An introductionto the Cheeger Problem II

## Gian Paolo Leonardi Università di Modena

Abstract: the Cheeger problem consists in minimizing the ratio between perimeter and volume among subsets of a given set $\Omega$. The infimum of this ratio is the Cheeger constant of $\Omega$, while minimizers are called Cheeger sets. Quite surprisingly, this variational problem turns out to be closely linked to a number of other relevant problems (eigenvalue estimates, capillarity, image segmentation techniques, max-flow/min-cut duality, landslide models). After introducing some essential concepts and tools from the theory of BV functions and finite perimeter sets, we shall review some classical as well as recent results on this topic.
All lectures will be delivered at the Seminar Room in the 1st floor of the Mathematics building.
Lecture 1. March 20, 12'00–13'30. Introduction. Essentials on BV functions and finite perimeter sets.
Lecture 2: March 21, 16'00–17'30. General properties of Cheeger sets. The two-dimensional case.
Lecture 3: March 22, 12'00–13'30 Links with prescribed mean curvature equation and capillarity.

# An introduction to the Cheeger problem I

## Gian Paolo Leonardi Università di Modena

Abstract: the Cheeger problem consists in minimizing the ratio between perimeter and volume among subsets of a given set $\Omega$. The infimum of this ratio is the Cheeger constant of $\Omega$, while minimizers are called Cheeger sets. Quite surprisingly, this variational problem turns out to be closely linked to a number of other relevant problems (eigenvalue estimates, capillarity, image segmentation techniques, max-flow/min-cut duality, landslide models). After introducing some essential concepts and tools from the theory of BV functions and finite perimeter sets, we shall review some classical as well as recent results on this topic.
All lectures will be delivered at the Seminar Room in the 1st floor of the Mathematics building.
Lecture 1. March 20, 12'00–13'30. Introduction. Essentials on BV functions and finite perimeter sets.
Lecture 2: March 21, 16'00–17'30. General properties of Cheeger sets. The two-dimensional case.
Lecture 3: March 22, 12'00–13'30 Links with prescribed mean curvature equation and capillarity.

Sala de conferencias, primera planta

# Stability inequalities for perimeter-minimizing clusters

## Gian Paolo Leonardi Università di Modena

We address the problem of quantitative stability for perimeter-minimizing clusters in $\mathbb{R}^n$ with multiple volume constraints (soap bubble clusters). Our aim is to show that the perimeter deficit controls a suitable power of an asymmetry functional. A first reduction to a sequence of $\Lambda$-minimizing clusters that converge to a given minimizer is accomplished through a "selection principle" that relies on the regularity theory for clusters. Several basic questions, like the optimality of power 2 (i.e. the existence of "quadratic deformations" for any minimizing cluster), the connection between global and infinitesimal stability, and the global parametric representation of $\Lambda$-minimizers that are sufficiently close to a given minimizer, are considered. In the planar case, we prove sharp stability inequalities for standard double bubbles. Some applications and open problems will be also discussed.

Seminario de Matemáticas, 1ª planta

# Gian Paolo Leonardi

## Università di Modena

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5
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