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HiCoS SIGNED

Higher Co-dimension Singularities: Minimal Surfaces and the Thin Obstacle Problem

Total Cost €

EC-Contrib. €

Partnership

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HiCoS project word cloud

Explore the words cloud of the HiCoS project. It provides you a very rough idea of what is the project "HiCoS" about.

framework boundary prominently investigation dimension central lower plateau unified rectifiable prominent analytical techniques final surfaces assigned boundedness dirichlet answering thin singularities suited minimizing hypersurfaces consisting contexts settings valuable compelling area dimensional regard currents conjectures examples geometric regularity consideration questions points variational mass principles variations mathematics suitable free underlines calculus objects theory tradition minimal naturally attacking functions differential space least unifying variant valid played regular equations partial physical insights co structure energy limited provides theme constrained solutions obstacle integer basis ubiquitous purpose singular content fruitful

Project "HiCoS" data sheet

The following table provides information about the project.

Coordinator	UNIVERSITA DEGLI STUDI DI ROMA LA SAPIENZA Organization address address: Piazzale Aldo Moro 5 city: ROMA postcode: 185 website: www.uniroma1.it contact info title: n.a. name: n.a. surname: n.a. function: n.a. email: n.a. telephone: n.a. fax: n.a.
Coordinator Country	Italy [IT]
Total cost	1˙341˙250 €
EC max contribution	1˙341˙250 € (100%)
Programme	1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
Code Call	ERC-2017-STG
Funding Scheme	ERC-STG
Starting year	2018
Duration (year-month-day)	from 2018-02-01 to 2023-01-31

Partnership

Take a look of project's partnership.

#	participants	country	role	EC contrib. [€]
1	UNIVERSITA DEGLI STUDI DI ROMA LA SAPIENZA UNIVERSITA DEGLI STUDI DI ROMA LA SAPIENZA Organization address address: Piazzale Aldo Moro 5 city: ROMA postcode: 185 website: www.uniroma1.it contact info title: n.a. name: n.a. surname: n.a. function: n.a. email: n.a. telephone: n.a. fax: n.a.	IT (ROMA)	coordinator	1˙341˙250.00
2	UNIVERSITAET LEIPZIG UNIVERSITAET LEIPZIG Organization address address: RITTERSTRASSE 26 city: LEIPZIG postcode: 4109 website: www.uni-Ieipzig.de contact info title: n.a. name: n.a. surname: n.a. function: n.a. email: n.a. telephone: n.a. fax: n.a.	DE (LEIPZIG)	participant	0.00

Map

Project objective

Singular solutions to variational problems and to partial differential equations are naturally ubiquitous in many contexts, and among these minimal surfaces theory and free boundary problems are two prominent examples both for their analytical content and their physical interest. A crucial aspect in this regard is the co-dimension of the objects under consideration: indeed, many of the analytical and geometric principles which are valid for minimal hypersurfaces or regular points of the free boundary do not apply to higher co-dimension surfaces or singular free boundary points.

The aim of this project is to investigate some of the most compelling questions about the singularities of two classical problems in the geometric calculus of variations in higher co-dimension:

I. Mass-minimizing integer rectifiable currents, i.e. solutions to the Plateau problem of finding the surfaces of least area, attacking specific conjectures about the structure of the singular set, most prominently the boundedness of its measure.

II. The thin obstacle problem, consisting in minimizing the Dirichlet energy (or a variant of it) among functions constrained above an obstacle that is assigned on a lower dimensional space, with the purpose of answering some of the main open questions on the singular free boundary points.

The main unifying theme of the project is the central role played by geometric measure theory, which underlines various common aspects of these two problems and makes them suited to be treated in an unified framework. Although these are classical questions with a long tradition, our knowledge about them is still limited and their investigation is among the most challenging issues in regularity theory. This is the central focus of the project, with the final goal to develop suitable analytical techniques that provides valuable insights on the mathematics at the basis of higher co-dimension singularities, eventually fruitful in other geometric and analytical settings.

Publications

List of publications.
year	authors and title	journal	last update
2018	Camillo De Lellis, Andrea Marchese, Emanuele Spadaro, Daniele Valtorta Rectifiability and upper Minkowski bounds for singularities of harmonic $Q$-valued maps published pages: 737-779, ISSN: 0010-2571, DOI: 10.4171/cmh/449	Commentarii Mathematici Helvetici 93/4	2019-10-03
2018	Matteo Focardi, Francesco Geraci, Emanuele Spadaro Quasi-Monotonicity Formulas for Classical Obstacle Problems with Sobolev Coefficients and Applications published pages: , ISSN: 0022-3239, DOI: 10.1007/s10957-018-1398-y	Journal of Optimization Theory and Applications	2019-10-03
2018	Matteo Focardi, Emanuele Spadaro On the Measure and the Structure of the Free Boundary of the Lower Dimensional Obstacle Problem published pages: 125-184, ISSN: 0003-9527, DOI: 10.1007/s00205-018-1242-4	Archive for Rational Mechanics and Analysis 230/1	2019-10-03

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The information about "HICOS" are provided by the European Opendata Portal: CORDIS opendata.