Un nuovo approccio alle disuguaglianze isoperimetriche quantitative

Gian Paolo Leonardi


We introduce a new variational method for studying geometric and functional inequalities with quantitative terms. In the context of isoperimetric-type inequalities, this method (called Selection Principle) is based on a penalization technique combined with the regularity theory of quasiminimizers of the perimeter functional. In this seminar we present the method and describe two remarkable applications. The rst one is a new proof of the sharp quantitative isoperimetric inequality in Rn. The second one is the proof of a conjecture posed by Hall about the optimal constant in the quantitative isoperimetric inequality in R2, in the small asymmetry regime.

Full Text:

PDF (Italiano)


F. J. Almgren, Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc., 4 (1976), pp. viii+199.

A. Alvino, V. Ferone, and C. Nitsch, A sharp isoperimetric inequality in the plane, J. Eur. Math. Soc. (JEMS), 13 (2011), pp. 185{206.

L. Ambrosio, Corso introduttivo alla teoria geometrica della misura ed alle superci minime, Appunti dei Corsi Tenuti da Docenti della Scuola. [Notes of Courses Given by Teachers at the School], Scuola Normale Superiore, Pisa, 1997.

L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 2000.

T. Bonnesen, Uber das isoperimetrische Dezit ebener Figuren, Math. Ann., 91 (1924), pp. 252{268.

Y. D. Burago and V. A. Zalgaller, Geometric inequalities, vol. 285 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinski, Springer Series in Soviet Mathematics.

S. Campi, Isoperimetric decit and convex plane sets of maximum translative discrepancy, Geom. Dedicata, 43 (1992), pp. 71-81.

M. Cicalese and G. P. Leonardi, A selection principle for the sharp quantitative isoperimetric inequality, preprint, (2010).

---- , Best constants for the isoperimetric inequality in quantitative form, preprint, (2011).

E. De Giorgi, Frontiere orientate di misura minima, Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960-61, Editrice Tecnico Scientica, Pisa, 1961.

L. C. Evans and R. F. Gariepy, Measure theory and ne properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.

A. Figalli, F. Maggi, and A. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities, Invent. Math., 182 (2010), pp. 167-211.

B. Fuglede, Stability in the isoperimetric problem, Bull. London Math. Soc., 18 (1986), pp. 599-605.

N. Fusco, F. Maggi, and A. Pratelli, The sharp quantitative isoperimetric inequality, Ann. of Math. (2), 168 (2008), pp. 941-980.

E. Giusti, Minimal surfaces and functions of bounded variation, vol. 80 of Monographs in Mathematics, Birkhauser Verlag, Basel, 1984.

R. R. Hall, A quantitative isoperimetric inequality in n-dimensional space, J. Reine Angew. Math., 428 (1992), pp. 161-176.

R. R. Hall and W. K. Hayman, A problem in the theory of subordination, J. Anal. Math., 60 (1993), pp. 99-111.

R. R. Hall, W. K. Hayman, and A. W. Weitsman, On asymmetry and capacity, J. Anal. Math., 56 (1991), pp. 87-123.

F. Maggi, Some methods for studying stability in isoperimetric type problems, Bull. Amer. Math. Soc. (N.S.), 45 (2008), pp. 367{408.

U. Massari, Esistenza e regolarita delle ipersuperci di curvatura media assegnata in Rn, Arch. Rational Mech. Anal., 55 (1974), pp. 357{382.

I. Tamanini, Boundaries of Caccioppoli sets with Holder-continuous normal vector, J. Reine Angew. Math., 334 (1982), pp. 27-39.

---- , Regularity results for almost minimal oriented hypersurfaces in Rn, Quaderni del Dipartimento di Matematica dell' Universita di Lecce, 1 (1984), pp. 1-92.

DOI: 10.6092/issn.2240-2829/2671


  • There are currently no refbacks.

Copyright (c) 2011 Gian Paolo Leonardi

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 Unported License.