Related papers: Symmetry of minimizers of a Gaussian isoperimetric…
In this paper, we investigate the minimization of a functional in which the usual perimeter is competing with a nonlocal singular term comparable (but not necessarily equal to) a fractional perimeter. The motivation for this problem is a…
We study a geometric variational problem for sets in the plane in which the perimeter and a regularized dipolar interaction compete under a mass constraint. In contrast to previously studied nonlocal isoperimetric problems, here the…
We consider the volume-constrained minimization of the sum of the perimeter and the Riesz potential. We add an external potential of the form $\|x\|^{\beta}$ that provides the existence of a minimizer for any volume constraint, and we study…
We consider a class of generalized antiferromagnetic local/nonlocal interaction functionals in general dimension, where a short range attractive term of perimeter type competes with a long range repulsive term characterized by a reflection…
In the main theorem of this paper we treat the problem of existence of minimizers of the isoperimetric problem under the assumption of small volumes. Applications of the main theorem to asymptotic expansions of the isoperimetric problem are…
We consider a non-local isoperimetric problem with a repulsive Coulombic term. In dimension three this corresponds to the Gamow's famous liquid drop model. We show that whenever the mass is small the ball is the unique minimizer of the…
Derived from the concentration-compactness principle, the concept of generalized minimizer can be used to define generalized solutions of variational problems which may have components ``infinitely'' distant from each other. In this article…
We study an isoperimetric problem the energy of which contains the perimeter of a set, Coulomb repulsion of the set with itself, and attraction of the set to a background nucleus as a point charge with charge $Z$. For the variational…
This paper is concerned with a study of the classical isoperimetric problem modified by an addition of a non-local repulsive term. We characterize existence, non-existence and radial symmetry of the minimizers as a function of mass in the…
We investigate, under a volume constraint and among sets contained in a Euclidean half-space, the minimization problem of an energy functional given by the sum of a capillarity perimeter, a nonlocal interaction term and a gravitational…
We characterize the volume-constrained minimizers of a nonlocal free energy given by the difference of the $t$-perimeter and the $s$-perimeter, with $s$ smaller than $t$. Exploiting the quantitative fractional isoperimetric inequality, we…
We prove a quantitative version of the isoperimetric inequality for a non local perimeter of Minkowski type. We also apply this result to study isoperimetric problems with repulsive interaction terms, under convexity constraints. We show…
Given a positive lower semi-continuous density $f$ on $\mathbb{R}^2$ the weighted volume $V_f:=f\mathscr{L}^2$ is defined on the $\mathscr{L}^2$-measurable sets in $\mathbb{R}^2$. The $f$-weighted perimeter of a set of finite perimeter $E$…
This paper presents a geometric approach to the classical isoperimetric problem by analysing the efficiency of regular polygons in enclosing maximum area for a fixed perimeter. Using efficiency metrics, it proves that regular polygons…
For $\Omega_\e=(0,\e)\times (0,1)$ a thin rectangle, we consider minimization of the two-dimensional nonlocal isoperimetric problem given by \[ \inf_u E^{\gamma}_{\Omega_\e}(u)\] where \[ E^{\gamma}_{\Omega_\e}(u):=…
We prove a relative isoperimetric inequality in the plane, when the perimeter is defined with respect to a convex, positively homogeneous function of degree one, and characterize the minimizers.
The existence of minimizers in the fractional isoperimetric problem with multiple volume constraints is proved, together with a partial regularity result.
We study the localization of sets with constant nonlocal mean curvature and prescribed small volume in a bounded open set with smooth boundary, proving that they are {\em sufficiently close} to critical points of a suitable non-local…
The search of the optimal constant for a generalized Wirtinger inequality in an interval consists in minimizing the $p$-norm of the derivative among all functions whose $q$-norm is equal to~1 and whose $(r-1)$-power has zero average.…
In this paper we study the functional given by the integral of the mean curvature of a convex set with Gaussian weight with Gaussian volume constraint. It was conjectured that the ball centered at the origin is the only minimizer of such a…