# Isoperimetry and Stability Properties of Balls with Respect to Nonlocal Energies

@article{Figalli2015IsoperimetryAS, title={Isoperimetry and Stability Properties of Balls with Respect to Nonlocal Energies}, author={Alessio Figalli and Nicola Fusco and Francesco Maggi and Vincent Millot and Massimiliano Morini}, journal={Communications in Mathematical Physics}, year={2015}, volume={336}, pages={441-507} }

We obtain a sharp quantitative isoperimetric inequality for nonlocal s-perimeters, uniform with respect to s bounded away from 0. This allows us to address local and global minimality properties of balls with respect to the volume-constrained minimization of a free energy consisting of a nonlocal s-perimeter plus a non-local repulsive interaction term. In the particular case s = 1, the s-perimeter coincides with the classical perimeter, and our results improve the ones of Knuepfer and Muratov…

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## References

SHOWING 1-10 OF 49 REFERENCES

### On an Isoperimetric Problem with a Competing Nonlocal Term II: The General Case

- Mathematics
- 2013

This paper is the continuation of a previous paper (H. Knüpfer and C. B. Muratov, Comm. Pure Appl. Math. 66 (2013), 1129–1162). We investigate the classical isoperimetric problem modified by an…

### LOCAL AND GLOBAL MINIMALITY ISSUES FOR A NONLOCAL ISOPERIMETRIC PROBLEM ON R N

- Mathematics
- 2013

. We consider a nonlocal isoperimetric problem deﬁned in the whole space R N , whose nonlocal part is given by a Riesz potential with exponent α ∈ (0 ,N − 1). We show that critical conﬁgurations with…

### Local and Global Minimality Results for a Nonlocal Isoperimetric Problem on ℝN

- MathematicsSIAM J. Math. Anal.
- 2014

It is shown that critical configurations with positive second variation are local minimizers and satisfy a quantitative inequality with respect to the $L^1$-norm.

### Minimality via Second Variation for a Nonlocal Isoperimetric Problem

- Mathematics
- 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…

### Uniform estimates and limiting arguments for nonlocal minimal surfaces

- Mathematics
- 2011

We consider nonlocal minimal surfaces obtained by a fractional type energy functional, parameterized by $${s\in(0,1)}$$. We show that the s-energy approaches the perimeter as s → 1−. We also provide…

### Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory

- Mathematics
- 2012

The marriage of analytic power to geometric intuition drives many of today's mathematical advances, yet books that build the connection from an elementary level remain scarce. This engaging…

### Stability in the isoperimetric problem for convex or nearly spherical domains in ⁿ

- Mathematics
- 1989

For convex bodies D in Rn the deviation d from spherical shape is estimated from above in terms of the (dimensionless) isoperimetric deficiency A of D as follows: d < f(A) (for A sufficiently small).…

### Regularity for non-local almost minimal boundaries and applications

- Mathematics
- 2010

We introduce a notion of non-local almost minimal boundaries similar to that introduced by Almgren in geometric measure theory. Extending methods developed recently for non-local minimal surfaces we…

### A Selection Principle for the Sharp Quantitative Isoperimetric Inequality

- Mathematics
- 2010

We introduce a new variational method for the study of isoperimetric inequalities with quantitative terms. The method is general as it relies on a penalization technique combined with the regularity…

### Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints

- Mathematics
- 1976

This is a research announcement of results [Al] the full details and proofs of which have been submitted for publication elsewhere. We study the structure of m dimensional subsets of R which are well…