Calculus of Variations and Geometric Measure Theory

E. Acerbi - N. Fusco - M. Morini

Minimality via second variation for a nonlocal isoperimetric problem

created by morini on 30 Jun 2011
modified by acerbi on 11 Apr 2024


Published Paper

Inserted: 30 jun 2011
Last Updated: 11 apr 2024

Journal: Communications in Mathematical Physics
Volume: 322
Pages: 515--557
Year: 2013


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.