Curriculum Vitae

Curriculum Vitae (English)
Quick links to: list of publications, research interests and teaching

Personal information:
- Born August 19 1978 in Viareggio, Italia;
- e-mail: maggi AT math.unifi.it;
- Tel. office: +39-055-4237-496;
- Languages: Italian, English;
- Hobbies: Guitar.

Positions:
- From September 2010 to May 2011 I'll be visiting the Institute for Computer Science and Engeneering and the Department of Mathematics of the University of Texas at Austin
- From December 23, 2011: tenured associate professor, Università di Firenze
- From October 1, 2005 to December 22, 2011: tenured assistant professor (ricercatore a tempo indeterminato), Università di Firenze
- From December 2004 to September 2005: wissenshaftlichen assistenten C3 with Sergio Conti, Universität Duisburg-Essen
- From May 2004 to December 2004 I have been a post doctoral associate at the Max Planck Institute For Mathematics In The Sciences in Leipzig, Germany. I had a grant in the project "Multiscale folding patterns in compressed elastic sheets" which leader was Sergio Conti. This project was part of the DFG Priority Program 1095 "Analysis, Modeling and Simulation of Multiscale Problems"

Formation and prizes:
- Habilitation as associate professor in Mathematical Analysis, August 2010.
- Premio Carlo Miranda 2008, Accademia di Scienze Fisiche e Matematiche di Napoli.
- From April 2001 until April 2004 I have been a PhD student in Mathematics at the Università di Firenze with Paolo Marcellini as advisor. The title of my PhD thesis is On Serrin's lower semicontinuity problem. I have defended it in September 2004.
- I have been a student in Mathematics at the Università di Firenze from September 1996 to July 2000, when I obtained my degree in Mathematics (Laurea) with the lode of the jury. Also in that occasion my advisor was Paolo Marcellini. I presented a thesis titled ``Funzioni quasiconvesse nel senso di Morrey''.

Visits:
- Institut für Angewandte Mathematik, Bonn, from 26.9.10 to 1.10.10
- Instituto de Ciencias Matemáticas, Madrid, from 19.9.10 to 24.10.10
 - Hausdorff Research Institute for Mathematics, Junior Trimester Program on Analysis Group "Calculus of Variations and Image Processing", from 31.8.08 to 5.9.08.
- Dario Cordero-Erausquin, Equipe d'Analyse Fonctionnelle, UMPC (Paris 6), from 12.12.07 to 13.12.07.
- Irene Fonseca and Giovanni Leoni at the Center of Nonlinear Analysis of Carnegie Mellon University, Pittsburgh, from 18.11.07 to 2.12.07.
- Aldo Pratelli at the Università di Pavia from 24.10.07 to 25.10.07.
- Sergio Conti at the Universität Duisburg-Essen from 24.6.07 to 28.6.07.
- Nicola Fusco at the Università di Napoli during May 2007.
- Aldo Pratelli at the Università di Pavia from 5.3.07 to 7.3.07.
- Enrico Valdinoci at the Università di Roma "Tor Vergata", from 28.2.07 to 2.3.07.
- Irene Fonseca and Giovanni Leoni at the Center of Nonlinear Analysis of Carnegie Mellon University, Pittsburgh, from 18.11.06 to 3.12.06.
- Antonio De Simone at SISSA, Trieste, from 28.03.06 to 31.03.06.
- Camillo De Lellis at the University of Zurich from 31.01.06 to 3.02.06.
- Nicola Fusco at the Università di Napoli from 6.04.05 to 8.04.05.
- Felix Otto at theInstitut für Angewandte Mathematik Abteilung für Mathematische Methoden der Physik, Bonn, from 19.07.04 to 23.07.04.
- I have visited twice Stefan Müller's research group at the MPI-MIS, the first time from October to December 2002 and the second time for the same period just one year after. During my first stay in Leipzig, I also spent some days in Lyon, at the Unité de Mathématiques Pures et Appliquées, Ecole Normale Supérieure de Lyon, invited by Cédric Villani.
- From January to May 2003 I have been at the Center for Nonlinear Analysis, Carnagie Mellon University of Pittsburgh, visiting Irene Fonseca.

Invited lectures:
- "Hoelderianità e semicontinuità inferiore in L^1" Workshop "Calcolo delle Variazioni e Teoria Geometrica della Misura", Levico Terme, February 2002.
- "Le palle hanno la peggior disuguaglianza di Sobolev ottimale" Workshop "Calcolo delle Variazioni e Teoria Geometrica della Misura", Levico Terme, February 2004.
- "A new approach to counterexamples to L^1 estimates: Korn's inequality and geometric rigidity" Workshop "Dislocation Patterns in Plastic Materials", Warwick, May 2004.
- "Sul problema della regolarità delle funzioni rango-uno convesse" Workshop “Calcolo delle Variazioni e Teoria Geoemetrica della Misura”, Lizzanello, Centro Polifunzionale Ennio De Giorgi,  October 2004.
- "Isometric embeddings and scaling laws for compressed elastic sheets" Workshop "Recent Advances in Calculus of Variations and PDE's: a young researchers meeting", Dip. Matematica Tonelli, Pisa, March 2005.
- "Confining thin elastic sheets and folding paper" Workshop "Mutiscale Problems in Quantuum Mechanics and Averaging Techniques", Weierstrass Institute, Berlin, September 2005.
- "The isoperimetric inequality in a sharp, quantitative form" Workshop "Optimal Transport and Geometric PDE's", Nice, June 2006.
- "A quantitative version of the isoperimetric inequality", joint contributed talk with Aldo Pratelli. Workshop "Calculus of Variations", MFO, Oberwolfach, July 2006.
- "The quantitative Sobolev inequality in the supercritical case" "Summer School on Calculus of Variations and Applications", Ponta Delgada, Açores, September 2006.
- "Una forma quantitativa della disuguaglianza di Sobolev" Workshop "Calcolo delle Variazioni e Teoria Geometrica della Misura", Levico Terme, February 2007.
- "A stability estimate for the first eigenvalue of the p-Laplacean" Workshop "New trends in PDEs and Calculus of Variations", Cortona, May 2007.
- "L'energia elastica necessaria a comprimere un foglio di carta" XVIII Congresso dell'Unione Matematica Italiana, September 2007.
- "Trasporto di massa e stabilità nel problema di Wulff" Workshop "XVIII Convegno Nazionale di Calcolo delle Variazioni", Levico Terme, February 2008.
- "Symmetrization, optimal transport and quantitative isoperimetric inequalities" Workshop "Glimpses of Geometry", ENS-Lyon, 15-17 May 2008.
- "Stability problems for crystals" Workshop "Advances in Mathematical Analysis", EPFL Lausanne, 26 March 2009.
- "Stability problems for anisotropic surface tensions" "Mini-symposium in PDEs",Maxwell Institute Centre for Analysis and Nonlinear PDEs Edinburgh, 5 May 2009.
- "Stability problems for anisotropic surface tensions" Workshop "Optimal transportation: theory and applications", Institut Fourier, Grenoble, July 2009.
- "Stability problems for anisotropic surface tensions" Workshop "Recent advances in optimal transportation and applications", Nice, October 2009.
- "An overview on sharp stability inequalities in isoperimetric problems" GNAMPA-ERC Summer school, Ischia, June 2010.
- "An isoperimetric inequality of mixed Euclidean-Gaussian type" Intensive trimester on "Calculus of Variations, Singular Integrals and Incompressible Flows", ICMAT, Madrid, September 2010
- “Sulla caratterizzazione dei minimi nei problemi isoperimetrici rispetto a densità simmetriche”, XXI Convegno nazionale di Calcolo delle Variazioni, Levico Terme, February 2011

Research seminars:
- "The common root of the geometric conditions in Serrin's lower semicontinuity theorem",  MFO, Oberwolfach, October 2002.
-"Transport of mass and sharp Sobolev inequalities with trace terms", Arbeitsgmeinschat Mikrostrukturen, MPI-MIS, Leipzig, November 2003.
- "Le sfere hanno la peggior disuguaglianza di Sobolev ottimale", Dipartimento di Matematica "Ulisse Dini", February and April 2004.
- "Isometric embeddings and scaling laws for compressed elastic sheets", Arbeitsgmeinschat Mikrostrukturen, MPI-MIS, Leipzig, January 2005.
- "L'energia elastica necessaria a comprimere un foglio di carta e questioni matematiche collegate", Dipartimento di Matematica Università di Napoli, April 2005.
- "The sharp quantitative isoperimetric inequality", Institut für Mathematik, Universität Zurich, February 2006.
- "Confining thin elastic sheets and folding paper", SISSA, Trieste, March 2006.
- "The sharp quantitative Sobolev inequality for functions of bounded variation", CNA, Carnagie Mellon University, Pittsburgh, November 21 2006.
- "Una versione quantitativa della disuguaglianza di Sobolev", Dipartimento di Matematica Univ. di Firenze, January 19, 2007.
- "Una forma quantitativa della disuguaglianza di Sobolev", Università di Roma "Tor Vergata", March 2007.
- "Stability properties of the Euclidean isoperimetric inequality", Universität Duisburg-Essen June 2007.
- "Stability properties of the Wulff shape via mass transportation", CNA, Carnagie Mellon University Pittsburgh, November 27 2007.
- "Stability properties of the Wulff shape via mass transportation", Séminaire d'Analyse Fonctionnelle UMPC (Paris 6), December 13 2007.
- "Una stima di stabilità per il problema isoperimetrico relativo a un perimetro anisotropo", Seminario del gruppo di ricerca CVGMT, Pisa, December 19, 2007.
- "Sulla forma dei  cristalli di massa piccola", Dipartimento di Matematica Univ. Napoli "Federico II", January 30, 2009.
- "Stability problems for anisotropic surface tensions", Dipartimento di Fisica Univ. Roma Tor Vergata, November 5, 2009.
- "Stability problems for anisotropic surface tensions", Dipartimento di Matematica Univ. di Ferrara, November 27, 2009.
- "The isoperimetric problem with respect to a symmetric density", Dipartimento di Matematica Univ. di Padova, January 21, 2011.
- "The isoperimetric problem with respect to a symmetric density", Dipartimento di Matematica Univ. di Roma la Sapienza, January 31, 2011.
- "The isoperimetric problem with respect to a symmetric density", Dipartimento di Matematica Univ. di Roma Tor Vergata, February 1, 2011.
- "Sharp stability estimates in geometric variational problems", Department of Mathematics, Univ. of Texas at Austin, October 26, 2011.
- "Sharp stability estimates for the Plateau problem", Analysis Applied Math Seminar Univ. of Toronto, November 16, 2011.
- "Sharp stability estimates for the Plateau problem", Math Colloquium, Carnegie Mellon University, December 7, 2011.
- "Sharp stability estimates in geometric variational problems", Erlangen-Nurnberg University, Germany, January 20, 2012.
- "Sharp stability estimates for the Plateau problem", University of Houston, Texas, USA, March 30, 2012.

Invited graduate or research-level courses:
- "Disuguaglianze geometrico-funzionali in forma ottimale e quantitativa". Università degli Studi di Napoli "Federico II"., PhD course, May 2007. This paper originated from the lecture notes of this course.
- "Symmetrization, optimal transport and quantitative isoperimetric inequalities". Short-course in the school "Optimal transportation, geometry and functional inequalities", SNS Pisa, October 28-31, 2008.
- "Equilibrium shapes for anisotropic surface energies". Crash-course in Analysis and Nonlinear PDEs, co-sponsored by the Maxwell Centre for Analysis and Nonlinear PDEs and the Oxford Centre for Nonlinear PDE. Edinburgh, February 11-13, 2010.

Conference organization:
- (with Aldo Pratelli) ERC School on "Calculus of Variations, Continuum Mechanics and Geometric Inequalities", Ischia, June 2011.

Research projects:
- "Analysis of optimal sets and optimal constants: old questions and new results" (2010-2015). This is an ERC-Starting Grant. The principal investigator is Aldo Pratelli. I join this project as (the only) co-investigator.
- "Analytic techniques for geometric and functional inequalities" (2009-2014). This is an ERC-Advanced Grant. The principal investigator is Nicola Fusco. I partecipate to the project as a collaborator and manage 5% of this fund.
- "Disuguaglianze geometrico-funzionali in forma ottimale e quantitativa" (2007-2008), funded by INdAM-GNAMPA. I have managed this project together with Aldo Pratelli.

 Last update: March 30, 2012

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Research interests and collaborations

My research interests are in the Calculus of Variations and Geometric Measure Theory. A complete list of the authors I have collaborated (or I am collaborating) with, is the following: Eric Carlen, Andrea Cianchi, Marco Cicalese, Sergio Conti, Guido De Philippis, Daniel Faraco, Alessio Figalli, Irene Fonseca, Nicola Fusco, Michele Gori, Gian Paolo Leonardi, Giovanni Leoni, Paolo Marcellini, Massimiliano Morini, Stefan Müller, Marcello Ponsiglione, Aldo Pratelli and Cédric Villani. Here it follows a brief description (not so updated) of our work.

  1. Lower semicontinuity theorems


Me and James Serrin

I have been working (together with Michele Gori, Nicola Fusco and Paolo Marcellini) on some questions arising from a 1961 lower semicontinuity theorem by James Serrin. This work is spread in the papers [1, 3, 4, 6] and is collected in a more systematic form in my Ph.D thesis (On Serrin's lower semicontinuity problem, advisor Paolo Marcellini).

  2. Sobolev-type inequalities in sharp form

In the framework of mass transportation techniques and in collaboration with Cédric Villani, I have studied Sobolev inequalities with trace terms, and their connection with the isoperimetric inequality [8]. In particular this work has led us to answer a 1986 question by Brezis and Lieb, and to merge into a same inequality both the sharp Sobolev inequality and the sharp isoperimetric inequality. In a companion paper [9], we have further developed these techniques to the study of certain Gagliardo-Nirenberg inequalities, logarithmic Sobolev inequalities and Faber-Krahn inequalities, all with trace terms and merging between their usual formulation on R^n and the sharp isoperimetric inequality. Similar results are obtained in critical Sobolev spaces.


A way to look at trace Sobolev inequalities

 3. Nonliner elasticity

In collaboration with Sergio Conti I have studied paper folding and crumpling in the context of non-linear Elasticity [12]. In the same area of research, Sergio Conti, Stefan Müller and I have studied some aspects of the "limit theory for vanishing thickness" of a thin elastic sheet which boundary is kept fixed [11]. 


Two ways of crumpling a sheet of paper

  4. Singular vector valued maps


A weird staircase: step one - horizontal spreading, step two - vertical splitting

Together with Sergio Conti, Daniel Faraco and Stefan Müller, I have explored a new technique for producing counterexamples to L^1 estimates on vector valued maps [7, 10]. This method originated from previous research work by Faraco, and seemed to be suitable to apply in quite different situations. Further examples (together with references to further examples too) can be found in the paper Convex integration and the $L^p$ theory of elliptic equations by Kari Astala, Daniel Faraco, and László Székelyhidi.

5. Quantitative geometric-functional inequalities


In collaboration with Nicola Fusco and Aldo Pratelli [13], a sharp quantitative version of the isoperimetric inequality, first conjectured by Hall, has been proved. The strategy adopted in the proof is based on the idea of gradually symmetrize the object under analysis, in order to reduce to more and more symmetric sets without any loss in the rate of control. These ideas turn out to have a broader range of application. In [14] we show how to prove a sharp quantitative version of the Sobolev embedding for functions of bounded variation. The stability for the Sobolev inequality on W^{1,p}(R^n), 1<p<n, is presented in a joint work with Andrea Cianchi, Nicola Fusco and Aldo Pratelli [15]. In [16], we address a quantitative study of isocapacitary and Faber-Krahn type inequalities, with applications to Cheeger constants as well. In [17] a quantitative version of the Guassian isoperimetric inequality is established. A survey on these techniques is available in the paper [19].

These results are based on the use of quantitative rearrangement inequalities. A different approach, based on mass transportation techniques, is adopted to establish a quantitative anisotropic isoperimetric inequality, leading to a stability study of the Wulff shape of an elastic crystal: this is a joint paper with Alessio Figalli and Aldo Pratelli [18].
 


Shifted half balls

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List of publications, preprints and papers in preparation:

Publications and submitted papers:

  1. On some sharp conditions for lower semicontinuity in L^1. Joint work with M. Gori and P. Marcellini, Differential and Integral Equations 16 (2003), no.1, 51-76.
  2. On the lower semicontinuity of supremal functionals. Joint work with Michele Gori, ESAIM: Control Optim. Calc. Var. 9, (2003), 135-143.
  3. The common root of the geometric conditions in Serrin's lower semicontinuity theorem. Joint work with Michele Gori, Ann. Mat. Pura Appl. (4) 184 (2005), no. 1, 95--114.
  4. On the relaxation on BV of certain non coercive integral functionals. J. Convex Anal. 10 (2003), no. 2, 477--489.
  5. A Gamma convergence result for variational integrators of lagrangians with quadratic growth. Joint work with Massimiliano Morini, ESAIM Control Optim. Calc. Var. 10 (2004), no. 4, 656--665 [preprint n.8, 2003, CNA-CMU].
  6. A remark on Serrin's theorem. Joint work with Michele Gori and Nicola Fusco, NoDEA: Nonlinear Differential Equations and Applications, 13 (2006), no.4, 425--433 [preprint n. 48, 2003, Dip. Matematica e Applicazioni ``R.Caccioppoli'' Univ. Napoli]. 
  7. A new approach to counterexamples to L^1 estimates: Korn's inequality, geometric rigidity and regularity for gradients of separately convex functions. Joint work with Sergio Conti and Daniel Faraco, Arch. Ration. Mech. Anal. 175 (2005), no. 2, 287--300 [preprint n. 93, 2003, MPI-MIS].
  8. Balls have the worst best Sobolev inequalities. Joint work with Cédric Villani, J. Geom. Anal. 15 (2005), no. 1, 83--121, [preprint n.32, 2004, MPI-MIS].
  9. Balls have the worst best Sobolev inequalities. Part two: variants and extensions.  Joint work with Cédric Villani , Calc. Var. Partial Differential Equations, 31 (2008), no. 1, 47-74,  [Preprint].
  10. Rank-one convex functions on 2x2 symmetric matrices and laminates on rank-three lines. Joint work with Sergio Conti, Daniel Faraco and Stefan Müller, Calc. Var. Partial Differential Equations 24 (2005), no. 4, 479-493, [preprint n.50, 2004, MPI-MIS].
  11. Rigorous derivation of Föppl's theory for clamped elastic membranes leads to relaxation. Joint work with Sergio Conti and Stefan Müller,SIAM J. Math. Anal. 38 (2006), no. 2, 657-680 [preprint n.54, 2005, MPI-MIS].
  12. Confining thin elastic sheets and folding paper. Joint work with Sergio Conti, Arch. Ration. Mech. Anal. 187 (2008), no. 1, 1-48, [preprint Universität Duisburg-Essen].
  13. The sharp quantitative isoperimetric inequality. Joint work with Nicola Fusco and Aldo Pratelli, Ann. of Math., 168 (2008), no. 3, 941-980, [preprint cvgmt.sns.it]. 
  14. The sharp quantitative Sobolev inequality for functions of bounded variation. Joint work with Nicola Fusco and Aldo Pratelli, J. Func. Anal., (2007) 244, n.1, 315-341, [preprint cvgmt.sns.it].
  15. The sharp Sobolev inequality in quantitative form. Joint work with Andrea Cianchi, Nicola Fusco and Aldo Pratelli, Journal of the European Mathematical Society, (2009) 11, n.5, 1105-1139 [preprint cvgmt.sns.it].
  16. Stability estimates for certain Faber-Krahn, isocapacitary and Cheeger inequalities. Joint work with Nicola Fusco and Aldo Pratelli, Annali Scuola Normale Superiore di Pisa, Vol. VIII, issue 1 (2009), 51-71. [preprint cvgmt.sns.it].
  17. On the isoperimetric deficit in the Gauss space. Joint work with Andrea Cianchi, Nicola Fusco and Aldo Pratelli, [preprint cvgmt.sns.it], accepted on American Journal of Mathematics.
  18. A mass transportation approach to quantitative isoperimetric inequalities. Joint work with Alessio Figalli and Aldo Pratelli, Inventiones Mathematicae, 182, Number 1, (2010), 167-211, [preprint cvgmt.sns.it].
  19. Some methods for studying stability in isoperimetric type problems. Bull. Amer. Math. Soc., 45 (2008), 367-408.
  20. A note on Cheeger sets. Joint work with Alessio Figalli and Aldo Pratelli, Proc. Amer. Math. Soc. 137 (2009), no. 6, 2057--2062. [preprint cvgmt.sns.it].
  21. A refined Brunn-Minkowski inequality for convex sets. Joint work with Alessio Figalli and Aldo Pratelli, Annales de l'Institut Henri Poincare (C) Non Linear Analysis Volume 26, Issue 6, November-December 2009, 2511-2519 [preprint cvgmt.sns.it]. 
  22. Exact reconstruction of damaged color images using a total variation model. Joint work with Irene Fonseca, Giovanni Leoni and Massimiliano Morini, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 27, Issue 5, September-October 2010, Pages 1291-1331 [preprint].
  23. On the shape of liquid drops and crystals in the small mass regime. Joint work with Alessio Figalli, Archive for Rational Mechanics and Analysis, Volume 201, Number 1, 143-207 [preprint cvgmt.sns.it].
  24. On the isoperimetric problem with respect to a mixed Euclidean-Gaussian density. Joint work with Nicola Fusco and Aldo Pratelli, Journal of Functional Analysis, Volume 260, Issue 12, 15 June 2011, Pages 3678-3717 [preprint cvgmt.sns.it]
  25. A geometric approach to correlation inequalities in the plane. Joint work with Alessio Figalli and Aldo Pratelli [preprint cvgmt.sns.it]
  26. Quantitative stability in the isodiametric inequality via the isoperimetric inequality. Joint work with Marcello Ponsiglione and Aldo Pratelli [preprint cvgmt.sns.it]
  27. Sharp stability theorems for the anisotropic Sobolev and log-Sobolev inequalities on functions of bounded variation (with A. Figalli and A. Pratelli) preprint.
  28. Sharp stability inequalities for the Plateau problem (with G. De Philippis), preprint.
  29. On the isoperimetric problem for radial log-convex densities (with A. Figalli), preprint.
  30. Quantitative improvements of the Brunn-Minkowski and Riesz inequalities and applications to the shape of droplets in non-local phase transitions (with E. Carlen), in preparation.
  31. Stability of minimal clusters and double-bubbles in the plane (with M. Cicalese and G. P. Leonardi), in preparation.

Books and lecture notes:

  1. Symmetrization, optimal transport and quantitative isoperimetric inequalities, inside Optimal transportation, Geometry and Functional inequalities. Proceedings of the school held in Pisa, October 2008. Edited by Luigi Ambrosio. Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series, 11. Edizioni della Normale, Pisa, 2010.
  2. Sets of finite perimeter and geometric variational problems: an introduction to Geometric Measure Theory, Cambridge Studies in Advanced Mathematics, 135, 2012, Cambridge University Press.

 Last update: February 24, 2012.

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Teaching: 

Students:

- Marco Caroccia (Master degree in Mathematics): Stime asintotiche per partizioni minimali del piano.
- Berardo Ruffini (Master degree in Mathematics): Riduzione al caso radiale per una versione quantitativa della disuguaglianza di Gagliardo-Nirenberg.


Courses:

2010/2011:  
Università degli studi di Firenze
- Calculus of Variations (for students in Mathematics).
- Calculus (for students in Biology, with Elvira Mascolo and Andrea Colesanti).

2009/2010:  
Università degli studi di Firenze
- Calculus of Variations (for students in Mathematics).
- Calculus (for students in Biology, with Gloria Papi).

2008/2009:  
Università degli studi di Firenze
- Calculus of Variations (for students in Mathematics).
- Calculus (for students in Biology, with Gloria Papi).

2007/2008:  
Università degli studi di Firenze
- Calculus of Variations (for students in Mathematics).
- Calculus (for students in Biology, with Gloria Papi).
- Calculus (for students in Chemestry, with Elena Comparini).

2006/2007:  
Università degli studi di Firenze.
- Calculus (for students in Biology, with Gloria Papi).
- Calculus (for students in Chemestry, with Elena Comparini).

2005/2006:  
Università degli studi di Firenze. 
- Calculus (for students in Biology, with Gloria Papi).
- Calculus (for students in Chemestry, with Elena Comparini).

2004/2005:  
Universität Duisburg-Essen.
- Exercise sessions for Nonlinear Functional Analysis. Theory was taught by Sergio Conti.
- Introduction to functions of bounded variation and sets of finite perimeter. I have given a self-contained presentation of some of the basic tools and ideas in geometric measure theory, starting from the construction of the Lebesgue measure to conclude with the proof of the classical De Giorgi's Rectifiability Theorem.

 Last update: August 12, 2010.

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