EPSRC Reference: 
EP/P007287/1 
Title: 
Symmetry of Minimisers in Calculus of Variations 
Principal Investigator: 
Cagnetti, Dr F 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Sch of Mathematical & Physical Sciences 
Organisation: 
University of Sussex 
Scheme: 
First Grant  Revised 2009 
Starts: 
01 April 2017 
Ends: 
31 July 2019 
Value (£): 
101,104

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
Calculus of variations studies problems which can be formulated in terms of a quantity that needs to be minimised. The isoperimetric problem falls into this category: among all the geometric shapes in the plane with the same area, which one has the smallest perimeter? This question is so natural and old that it was referred to even by Virgil in the Aeneid, in the 1st century BC. Despite this fact, a satisfactory and mathematically sound solution to this problem was found only in 1958, when Ennio De Giorgi proved that there is just one geometric shape with this property: the disk. A crucial step in De Giorgi's argument consists in showing that the solution of the problem exhibits certain symmetries. Indeed, once we are given this piece of information, we can focus on a very restricted class of geometric shapes, and this easily allows to conclude the proof. More generally, a fundamental question arising in many problems in calculus of variations is the following:
QUESTION 1: Are minimisers symmetric (in some sense)?
The first goal of this project is to answer the question above for some important variational problems related to isoperimetric and functional inequalities. To this aim, we will take advantage of the successful approach recently adopted by the PI and his collaborators in the study of Steiner's and Ehrhard's inequality, where necessary and sufficient conditions for the symmetry of minimisers are given, in terms of the new notion of "essential connectedness". This notion explains what it means for a Borel set to disconnect another Borel set, and turns out to be crucial for a clear understanding of the aforementioned problems.
Once the symmetry of minimisers has carefully been investigated, a second important question arises:
QUESTION 2: Is it possible to "measure" in some way the asymmetry of those competitors that are "almost" minimisers?
For the isoperimetric problem, this second question can be formulated in the following way: suppose there is a geometric shape in the plane, whose perimeter is "close" to the perimeter of the disk. Can we say that such geometric shape is "almost" a disk? In 2008 Nicola Fusco, Francesco Maggi, and Aldo Pratelli gave a positive answer to this question, proving what is now referred to as the "quantitative isoperimetric inequality". This gives a precise estimate of how far from being a disk a geometric shape is, in terms of its perimeter. In recent years, quantitative inequalities have attracted the attention of a large number of mathematicians. This is because such inequalities not only imply stability of minimisers, but also give detailed nformation about competitors whose energy is close to the minimum value.
The second goal of this project is to prove, in the same spirit of what was done by Fusco, Maggi and Pratelli, a quantitative version of important isoperimetric and functional inequalities. These include, for instance, PólyaSzegö type inequalities, which have applications in several areas of mathematical analysis, such as harmonic analysis, spectral theory of differential operators, theory of Sobolev spaces, and PDEs.

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Organisation Website: 
http://www.sussex.ac.uk 