Free Discontinuity Problems

herausgegeben von: Nicola Fusco, Aldo Pratelli

Verlag: Scuola Normale Superiore

Buchreihe : CRM Series

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Inhaltsverzeichnis

Frontmatter

Fine regularity results for Mumford-Shah minimizers: porosity, higher integrability and the Mumford-Shah conjecture

Abstract

We review some classical results and more recent insights about the regularity theory for local minimizers of the Mumford and Shah energy and their connections with the Mumford and Shah conjecture. We discuss in details the links among the latter, the porosity of the jump set and the higher integrability of the approximate gradient. In particular, higher integrability turns out to be related with an explicit estimate on the Hausdorff dimension of the singular set and an energetic characterization of the conjecture itself.

Matteo Focardi

Variational models for epitaxial growth

Abstract

These lecture notes are based on a series of lectures that I gave at the ERC School on Free Discontinuity Problems, at the De Giorgi Center, Pisa, Italy, July 7–July 11, 2014. The school was organized by Aldo Pratelli and Nicola Fusco. Although the basic structure of the lecture notes follows the one given at the school, I am adding here more material and a lot of the details that I skipped in class.

Giovanni Leoni

Local and global minimality results for an isoperimetric problem with long-range interactions

Abstract

In this paper we review some recent results concerning the following non-local isoperimetric problem:

https://static-content.springer.com/image/chp%3A10.1007%2F978-88-7642-593-6_3/978-88-7642-593-6_3_Equa_HTML.gif

where m ∈ (-1,1) is given and prescribes the volume of the two phases {u = 1} and {u = -1}. Here P_Ω stands for the perimeter relative to Ω (in the sense of Caccioppoli-De Giorgi) and BV(Ω; {-1, 1}) denotes the space of functions of bounded variation taking values in {-1, 1}. We refer to [4] and [33] for a rather complete account on the properties of functions of bounded variations and sets of finite perimeter.

Massimiliano Morini

Backmatter

Titel: Free Discontinuity Problems
herausgegeben von: Nicola Fusco
Aldo Pratelli
Verlag: Scuola Normale Superiore
Electronic ISBN: 978-88-7642-593-6
Print ISBN: 978-88-7642-592-9
DOI: https://doi.org/10.1007/978-88-7642-593-6