Degree granting institution
School / College / Campus
On the isoperimetric properties of Planar N-clusters
Análise Matemática e Cálculo de Variação
|01-09-2015 - 01-08-2017||Post-doc fellowship.||Carnegie Mellon University|
Area of scientific activity
Area of scientific activity
My Ph.D thesis was about the asymptotic behavior of the energy of perimeter minimizing clusters in the plane. A stability result for the isoperitric honeycomb theorem and a sort of Cheeger’s problem for partitions involving Caffarelli and Lin’s conjecture about the optimal partitions for Laplacian eigenvalues are also studied. I am currently working at the Carnegie Mellon University, (Pittsburgh - PA) in the Center for Nonlinear Analysis under the supervision of Professor Irene Fonseca and Professor Giovanni Leoni. We are studying a surface evolution model involving the presence of adatoms, highly connected to materials science.
In my Ph.D thesis I have studied some isoperimetric problems involving planar tilings and Hales’ theorem about the so-called ”hexagonal honeycomb conjecture”. With my advisor we provide a result (in preparation) about the uniform distribution of the energy for the solutions of the isoperimetric equal area N-partition of a planar domain. The Theorem obtained is of the type of the one provided by Alberti, Choksi, Otto in for a kind of non-local isoperimetric problem in the plane.
Current main scientific area
I recently have had the chance to work on mathematical physics, material sciences and nonlinear analysis. In particular during Fall 2015 - Spring 2016 at the CMU I have started a still ongoing collarboration with Riccardo Cristoferi and Laurent Dietrich, under the supervision of Prof. Irene Fonseca and Giovanni Leoni. We started to work on the existence of solutions for a system of PDEs modeling the growth of a material under the presence of free adatoms moving on the top layer. Many author dedicated their interest to these kind of problems involving material sciences, starting from the pioneeristic work of Cahn and Taylor where different surface motion by mean curvature flow are analyzed from a gradient flow point of view, to the more recent contribution of Fonseca and Leoni. Our framework mainly refers to the work of Burger where the variational problem leading to the PDEs and driving the entire process is stated.
Other scientific activities
For a short period I have also focused my attention on the optimal transportation problems, in order to learn the techniques and the main issues in this topic, and on the Gamma-convergence approach for the treatment of some variational problems. My interests are oriented also in spectral optimization problems and I would like to enter in deep into the details of shape optimization. A previous work on these topics is obtained in collaboration with the Ph.D student Robin Neumayer in Austin. In the paper we provide a result involving a sharp quantitative version of the Cheeger inequality for poligons proved by Bucur and Fragala.
Participations in R&D projects
May 16, May 20, 2016 - New challenge for the calculus of variation stemming for problems in the material sciences and image processing - Conference, Montre´al, Quebec (CA)), Centre de recherches mathe´matiques (CRM), Universite´ de Montre´al, PO Box 6128, Station Centre-ville, Montre´al.
Workshop in honor of the 60th birthday of Irene Fonseca
Apr 11, Apr 15, 2016 - Microstructure Evolution in Materials: Defects, Cracks & Interfaces
Workshop, Leiden (NE), University of Leiden, Lorentz center, 3rd Floor Oort Building, Niels Bohrweg, Leiden (NE). - Workshop on materials science.
Dec 07, Dec 10, 2015 - SIAM Conference, Conference, Phoenix (AR), Double tree resort Hilton, Paradise Valley, Scottsdale, Phoenix (AR), US. Analysis of partial differential equations.
Oct 23, Oct 25, 2015 - IMA Special Workshop Workshop, Eugene (OR), Valley River Inn Hotel, Eugene (OR), US. Mathematics and Mechanics in the 22nd Century: Seven Decades and Counting (conference for the 90th birthday of Jerry Ericksen.)
Feb 02, Feb 06, 2015 - XXV National conference in Calculus of Variation - Conference, Levico Terme, Bellavista Relax Hotel, Viale Vittorio Emanuele III 7, 38056 (Tn), Trento, Italy. In february 2015 I attended a conference on calculus of variation organized by CIRM.
Aug 07, Dec 18, 2014 - Fall Semester in Austin - Courses and research, UT Texas at Austin, 2515 Speedway, 78712 Texas, (512) 471-7711, Austin, Texas (US).
I spent five months in Austin to work with my co advisor Francesco Maggi. We worked on some stability isoperimetric issues involving the hexagonal tiling in the plane. I also attended the course: Geometric Measure Theory (Teacher Francesco Maggi).
Jan 26, Jan 31, 2014 - XXIV National conference in Calculus of Variation - Conference, Levico Terme, Bellavista Relax Hotel, Viale Vittorio Emanuele III 7, 38056 (Tn), Trento, Italy.
In January 2014 I attended a conference on calculus of variation organized by CIRM.
Jul 15, Aug 03, 2014 - School and workshop on "geometric measure theory and optimal transport"
School and workshop, ICTP, Strada costiera 11, 3415 (Ts), Trieste, Italy.
The main topics of the school were the optimal transport theory and the theory of currents. The mini course on "optimal transport theory" were taught by Alessio Figalli and Guido De Philippis. As far as the course "theory of currents" is concerned the teachers were Camillo De Lellis and Emanuele Spadaro. At the end of the school I attend a one-week conference on these fields. The travel and the accomodation costs were covered by ICTP.
May 14, May 16, 2013 - Convexite´ cache´e en e´quations aux derive´es partielles non-line´aires
Lectures series, Universite´ catholique de Louvain, Institut de recherche en mathe´- matique et physique, Louvain La Neuve, Belgium.
Feb 03, Feb 08, 2013 - XXIII National conference in Calculus of Variation - Conference, Levico Terme, Bellavista Relax Hotel, Viale Vittorio Emanuele III 7, 38056 (Tn), Trento, Italy.
In february 2013 I attended a conference on calculus of variation organized by CIRM. The travel and accomodation costs were covered by CIRM.
Jan 28, Feb 01, 2013 - International Conference on Fluids And Variational Methods - Conference, University of Leipzig, Felix-Klein-Ho¨rsaal, Room 501, Paulinum, Uni- versity of Leipzig, Universita¨tsstraße 1, 04109, Leipzig, Germany.
In january 2013 I attended a conference about fluids and variational methods organized by Bernd Kirchheim, Felix Otto and La´szlo´ Sze´kelyhidi. The travel and accomodation costs were covered by Max planck institute.
Feb 05, Feb 10, 2012 - XXII National conference in Calculus of variation - Conference, Levico Terme, Bellavista Relax Hotel, Viale Vittorio Emanuele III 7, 38056 (Tn), Trento, Italy.
In february 2012 I have attended a conference on calculcus of variation organized by CIRM. The travel and accomodation costs were covered by CIRM.
Jul 31, Sep 2, 2011 - Summer School, Universita´ degli studi di Perugia, Via Vanvitelli 1, 06123, Perugia, Italy. In August 2011 I attended the "SMI", (Scuola Matematica interuniversitaria) in Perugia. I at- tended two lectures in analysis named Partial Differential Equation (given by prof. Pagani, Carlo and Johnson, Russell) and Functional Analysis (given by prof. Milmann, Vitali and Eidelman, Yuli). The travel and accomodation costs were covered by University of Perugia.
Jun 14, Jun 22, 2010 - International Modelling Week - School and workshop, Universidad Complutense de Madrid, Plaza de Ciencias, 3 Ciudad Universitaria 28040, Madrid, Spain. Injune2010Iattendedthe"InternationalModellingWeek"inMadridofferedbytheUniversidad Complutense de Madrid trough a scolarship. The work, named "Calibration of single-factor HJM models of interest rates", was about a mathematical-economic model. Supervisors: Prof.
Gerardo Oleaga (UCM), Miguel Carillon Alvarez (Banco Santander).
Apr 15, 2016 - Seminar, University of Leiden, Lorentz Center, Leiden, Netherlands. Seminar, University of Cologne, Mathematical Institute, Ko¨ln, Germany.
Jan 8, 2016 - Seminar, Carnegie Mellon University, Center for Non Linear Analysis, Pittsburgh (PA), USA.
Jul 17 - 2015 - Seminar, Max Planck Institute, Max-Planck-Institute for mathematics in the sciences, Leipzig, Germany.
May 20, 2015 - Seminar, Universita` degli studi di Modena, Dipartimento di Scienze Fisiche, Infor- matiche, Matematiche, Modena, Italy.
- The Thesis aims to highlight some isoperimetric questions involving the, so-called, N-clusters. We first briefly recall the theoretical framework we are adopting. This is done in Chapter one. In chapter two we focus on the standard isoperimetric problem for planar N-cluster for large values of N and we provide an equidistribution energy-type results under some suitable assumption. The third Chapter is devoted to a stability results of the hexagonal honeycomb tiling. Finally in the fourth Chapter we consider a generalization of the Cheeger constant, defined as a minimization of a suitable energy among the class of the N-clusters. We show how this problem is related to the optimal partition problem for the first Dirichlet eigenvalue of the Laplacian introduced by Caffarelli and Fang-Hua Lin in 2007. We conclude, in Chapter five, with some remarks and some possible future direction of investigation.
Caroccia. On the isoperimetric properties of Planar N-clusters. Preprint arXiv:1601.07116
Artigos em revistas de circulação internacional com arbitragem científica
- We prove a sharp quantitative version of Hales' isoperimetric honeycomb theorem by exploiting a quantitative isoperimetric inequality for polygons and an improved convergence theorem for planar bubble clusters. Further applications include the description of isoperimetric tilings of the torus with respect to almost unit-area constraints or with respect to almost flat Riemannian metrics.
Marco Caroccia and Francesco Maggi. A sharp quantitative version of hales’ isoperimetric honeycomb theorem. Journal de Mathematiques Pures et Appliquees, 2016.
- In this paper we introduce a Cheeger-type constant defined as a minimization of a suitable functional among all the N-clusters contained in an open bounded set O. Here with N-Cluster we mean a family of N sets of finite perimeter, disjoint up to a set of null Lebesgue measure. We call any N-cluster attaining such a minimum a Cheeger N-cluster. Our purpose is to provide a non trivial lower bound on the optimal partition problem for the first Dirichlet eigenvalue of the Laplacian. Here we discuss the regularity of Cheeger N-clusters in a general ambient space dimension and we give a precise description of their structure in the the planar case. The last part is devoted to the relation between the functional introduced here (namely the N-Cheeger constant), the partition problem for the first Dirichlet eigenvalue of the Laplacian and the Caffarelli and Lin's conjecture.
Marco Caroccia. Cheeger n-clusters. arXiv preprint arXiv:1501.05923, 2015.
Artigos em revistas de circulação internacional com arbitragem científica
- The regular N-gon provides the minimal Cheeger constant in the class of all N-gons with fixed volume. This result is due to a work of Bucur and Fragalà in 2014. In this note, we address the stability of their result in terms of the L1 distance between sets. Furthermore, we provide a stability inequality in terms of the Hausdorff distance between the boundaries of sets in the class of polygons having uniformly bounded diameter. Finally, we show that our results are sharp, both in the exponent of decay and in the notion of distance between sets.
Caroccia, Neumayer. A note on the stability of the Cheeger constant of N-gons. Journal of Convex Analysis, Volume 22, Number 4, pgg. 1207- 1213 (2015)