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# 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

# 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 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.

# Geometric aspects of semilinear elliptic PDEs and minimal hypersurfaces on closed manifolds.

## Marco A.M. Guaraco University of Chicago

In this talk I will discuss both local and global properties of the stationary Allen-Cahn equation in closed manifolds. This equation, arising from the theory of phase transitions, has a strong connection with the theory of minimal hypersurfaces. I will summarize recent results regarding the analogy between both theories, focusing on min-max constructions. In particular, new insights into both Almgren-Pitts and Marques-Neves existence theories of minimal hypersurfaces will be discussed.

Seminario 1ª Planta, IEMATH

# Eddygledson Souza Gama

Despacho: IEMATH, B-8C

# Gabriel Ruiz-Hernández

## Instituto de Matemáticas, Universidad Nacional Autónoma de México

Despacho: D4, IEMath

# Marcos Paulo Tassi

## Universidade Federal de São Carlos

Despacho: IEMath B8-B

Despacho: IEMath

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