1 Dipartimento di Matematica ed Informatica “U. Dini”, Università di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy.
2 Normandie Univ, France; ULH, LMAH, FR CNRS 3335, 25 rue Philippe Lebon, 76600 Le Havre, France.
3 Institut Élie Cartan de Lorraine UMR CNRS 7502, Université de Lorraine, BP 70239, 54506 Vandoeuvre-les-Nancy cedex, France.
Received: 28 July 2015
Accepted: 27 December 2015
In this paper we study the quantitative isoperimetric inequality in the plane. We prove the existence of a set Ω, different from a ball, which minimizes the ratio δ(Ω) /λ2(Ω), where δ is the isoperimetric deficit and λ the Fraenkel asymmetry, giving a new proof of the quantitative isoperimetric inequality. Some new properties of the optimal set are also shown.
Mathematics Subject Classification: 28A75 / 49J45 / 49J53 / 49Q10 / 49Q20
Key words: Isoperimetric inequality / quantitative isoperimetric inequality / isoperimetric deficit / Fraenkel asymmetry / rearrangement / shape derivative / optimality conditions
We thank B. Kawohl for very useful discussions on the topic of this paper, in particular we thank him for having suggested the idea of Proposition 2.11.
This work started while CB was at the Institut Elie Cartan Nancy supported by the ANR CNRS project GAOS (Geometric Analysis of Optimal Shapes), and the research group INRIA CORIDA (Contrôle robuste infini-dimensionnel et applications). The research of CB is supported by the Fir Project 2013 “Geometrical and Qualitative Aspects of PDEs”. The work of GC was partially done during her “délégation CNRS” at University of Lorraine. AH is supported by the project ANR-12-BS01-0007-01-OPTIFORM Optimisation de formes financed by the French Agence Nationale de la Recherche (ANR). The three authors have been supported by the Fir Project 2013 “Geometrical and Qualitative Aspects of PDEs” in their visitings. All these institutions are gratefully acknowledged.
© EDP Sciences, SMAI 2017