Final Report Summary - ANTEGEFI (Analytic Techniques for Geometric and Functional Inequalities)
After the pioneering results of Bonnesen (1924), Fuglede (1989) and Hall (1992) the study of the quantitative versions of the isoperimetric inequality and of related geometric and functional inequalities has received a new impulse from the papers by Fusco, Maggi, Pratelli (Annals of Math., 2008) and by Figalli, Maggi, Pratelli (Invent. Math., 2010).
The main objective of this project was to investigate more deeply this topic and to develop it into a mature field of mathematical research. The results obtained so far show that this objective has been fully achieved and among the main results one should mention in particular the following two:
- the new proof given by Cicalese and Leonardi (Arch. Rat. Mech. Anal., 2012) of the quantitative isoperimetric inequality based on a suitable penalization and selection argument and on the use of the regularity theory of minimal surfaces; this proof not only shed a new light on the subject, but also led to the solution of a conjecture by Hall on the best constant in the quantitative isoperimetric inequality in R^2 in the regime of small Fraenkel asymmetry and to several applications to other stability issues such as the stability of the first eigenvalue of the laplacian with the sharp exponent proved by Brasco, De Philippis and Velichov (Duke Math. J., to appear) and the stability of Almgren’s isoperimetric inequality and of the isoperimetric inequality on the sphere proved by Bögelein, Duzaar and Fusco (preprints 2012 and 2013, respectively)
- the proof of the quantitative Gaussian isoperimetric inequality given by Cianchi, Fusco, Maggi and Pratelli (American J. of Math., 2013); together with this result they have also obtained a different proof of the property that half spaces are the only isoperimetric sets. Here the main argument is in the spirit of the one used in the euclidean case by De Giorgi and simplifies the proof first obtained in 2001 by Carlen and Kerce.
The project team has also obtained several quantitative versions of well known geometric inequalities, such as the Brunn-Minkowski, the isodiametric and the isoperimetric inequality for Steiner symmetrization, and of other functional inequalities such as the Krahn-Szegö and the Szegö-Weinberger inequalities concerning the second eigenvalue of the laplacian under boundary Dirichlet and Neumann conditions, respectively.
Finally, we have also proved an old conjecture by Auerbach (Fusco and Pratelli, J. Eur. Math. Soc., 2011), stating that among all Zindler sets, i.e. planar convex sets whose bisecting chords have all the same length, the so called Auerbach triangle is the only one minimizing the area and maximizing the perimeter.
The techniques developed to get all these results have been also successfully exploited to deal with various variational problems involving volume integrals and geometric quantities such as the perimeter.
Among these applications one should mention:
- a result by Figalli and Maggi (Arch. Rat. Mech. Anal., 2011) concerning the behavior of liquid drops or crystals lying in equilibrium under the action of a potential energy; in their paper they give a partial answer to some difficult questions raised by Almgren and Taylor
- a result by Acerbi, Fusco and Morini (Commun. Math. Phys., 2013), who prove the (quantitative) local minimality of certain critical configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts via a second variation approach.
One should also add that at the time the project started there were only few scattered results on this subject, all of them proved by ad hoc arguments, but no general tools or strategies. After five years the situation is completely different and we have now clearly outlined four different approaches: symmetrization, mass transportation, penalization and second variation. With these tools in hand we will certainly achieve other important objectives in this field and develop it further.
The main objective of this project was to investigate more deeply this topic and to develop it into a mature field of mathematical research. The results obtained so far show that this objective has been fully achieved and among the main results one should mention in particular the following two:
- the new proof given by Cicalese and Leonardi (Arch. Rat. Mech. Anal., 2012) of the quantitative isoperimetric inequality based on a suitable penalization and selection argument and on the use of the regularity theory of minimal surfaces; this proof not only shed a new light on the subject, but also led to the solution of a conjecture by Hall on the best constant in the quantitative isoperimetric inequality in R^2 in the regime of small Fraenkel asymmetry and to several applications to other stability issues such as the stability of the first eigenvalue of the laplacian with the sharp exponent proved by Brasco, De Philippis and Velichov (Duke Math. J., to appear) and the stability of Almgren’s isoperimetric inequality and of the isoperimetric inequality on the sphere proved by Bögelein, Duzaar and Fusco (preprints 2012 and 2013, respectively)
- the proof of the quantitative Gaussian isoperimetric inequality given by Cianchi, Fusco, Maggi and Pratelli (American J. of Math., 2013); together with this result they have also obtained a different proof of the property that half spaces are the only isoperimetric sets. Here the main argument is in the spirit of the one used in the euclidean case by De Giorgi and simplifies the proof first obtained in 2001 by Carlen and Kerce.
The project team has also obtained several quantitative versions of well known geometric inequalities, such as the Brunn-Minkowski, the isodiametric and the isoperimetric inequality for Steiner symmetrization, and of other functional inequalities such as the Krahn-Szegö and the Szegö-Weinberger inequalities concerning the second eigenvalue of the laplacian under boundary Dirichlet and Neumann conditions, respectively.
Finally, we have also proved an old conjecture by Auerbach (Fusco and Pratelli, J. Eur. Math. Soc., 2011), stating that among all Zindler sets, i.e. planar convex sets whose bisecting chords have all the same length, the so called Auerbach triangle is the only one minimizing the area and maximizing the perimeter.
The techniques developed to get all these results have been also successfully exploited to deal with various variational problems involving volume integrals and geometric quantities such as the perimeter.
Among these applications one should mention:
- a result by Figalli and Maggi (Arch. Rat. Mech. Anal., 2011) concerning the behavior of liquid drops or crystals lying in equilibrium under the action of a potential energy; in their paper they give a partial answer to some difficult questions raised by Almgren and Taylor
- a result by Acerbi, Fusco and Morini (Commun. Math. Phys., 2013), who prove the (quantitative) local minimality of certain critical configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts via a second variation approach.
One should also add that at the time the project started there were only few scattered results on this subject, all of them proved by ad hoc arguments, but no general tools or strategies. After five years the situation is completely different and we have now clearly outlined four different approaches: symmetrization, mass transportation, penalization and second variation. With these tools in hand we will certainly achieve other important objectives in this field and develop it further.