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Minimality via Second Variation for a Nonlocal Isoperimetric Problem

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Abstract

We discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts. We show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and, in fact, satisfy a quantitative isoperimetric inequality with respect to sets that are L 1-close. The link with local minimizers for the diffuse-interface Ohta-Kawasaki energy is also discussed. As a byproduct of the quantitative estimate, we get new results concerning periodic local minimizers of the area functional and a proof, via second variation, of the sharp quantitative isoperimetric inequality in the standard Euclidean case. As a further application, we address the global and local minimality of certain lamellar configurations.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università Degli Studi di Parma, Parma, Italy

    E. Acerbi & M. Morini

  2. Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università Degli Studi di Napoli “Federico II”, Napoli, Italy

    N. Fusco

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  1. E. Acerbi
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  2. N. Fusco
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  3. M. Morini
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Correspondence to N. Fusco.

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Communicated by P. Constantin

Dedicated to Sergio Spagnolo on his 70th birthday

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Acerbi, E., Fusco, N. & Morini, M. Minimality via Second Variation for a Nonlocal Isoperimetric Problem. Commun. Math. Phys. 322, 515–557 (2013). https://doi.org/10.1007/s00220-013-1733-y

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