scholarly journals Local and global minimality issues for a nonlocal isoperimetric problem on $\mathbb R^N$

2016 ◽  
Vol 27 (1) ◽  
pp. 37-50
Author(s):  
Marco Bonacini ◽  
Riccardo Cristoferi
Keyword(s):  
Isoperimetric Problem ◽  
Global Minimality

An isoperimetric problem with a Coulombic repulsion and attractive term

2019 ◽  
Vol 25 ◽  
pp. 14
Author(s):  
Domenico Angelo La Manna
Keyword(s):  
Point Charge ◽  
Isoperimetric Problem ◽  
Coulomb Repulsion ◽  
Local Minimizer ◽  
Coulombic Repulsion ◽  
Global Minimality ◽  
Optimal Radius

We study an energy given by the sum of the perimeter of a set, a Coulomb repulsion term of the set with itself and an attraction term of the set to a point charge. We prove that there exists an optimal radius r0 such that if r < r0 the ball Br is a local minimizer with respect to any other set with same measure. The global minimality of balls is also addressed.

scholarly journals Pansu–Wulff shapes in ℍ1

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Julián Pozuelo ◽  
Manuel Ritoré
Keyword(s):  
Mean Curvature ◽  
Isoperimetric Problem ◽  
Variational Formula ◽  
Geodesic Curvature ◽  
Vertical Axis ◽  
Singular Part ◽  
Tangent Plane ◽  
Curvature Function ◽  
Invariant Norm ◽  
The First Variation

Abstract We consider an asymmetric left-invariant norm ∥ ⋅ ∥ K {\|\cdot\|_{K}} in the first Heisenberg group ℍ 1 {\mathbb{H}^{1}} induced by a convex body K ⊂ ℝ 2 {K\subset\mathbb{R}^{2}} containing the origin in its interior. Associated to ∥ ⋅ ∥ K {\|\cdot\|_{K}} there is a perimeter functional, that coincides with the classical sub-Riemannian perimeter in case K is the closed unit disk centered at the origin of ℝ 2 {{\mathbb{R}}^{2}} . Under the assumption that K has C 2 {C^{2}} boundary with strictly positive geodesic curvature we compute the first variation formula of perimeter for sets with C 2 {C^{2}} boundary. The localization of the variational formula in the non-singular part of the boundary, composed of the points where the tangent plane is not horizontal, allows us to define a mean curvature function H K {H_{K}} out of the singular set. In the case of non-vanishing mean curvature, the condition that H K {H_{K}} be constant implies that the non-singular portion of the boundary is foliated by horizontal liftings of translations of ∂ ⁡ K {\partial K} dilated by a factor of 1 H K {\frac{1}{H_{K}}} . Based on this we can define a sphere 𝕊 K {\mathbb{S}_{K}} with constant mean curvature 1 by considering the union of all horizontal liftings of ∂ ⁡ K {\partial K} starting from ( 0 , 0 , 0 ) {(0,0,0)} until they meet again in a point of the vertical axis. We give some geometric properties of this sphere and, moreover, we prove that, up to non-homogeneous dilations and left-translations, they are the only solutions of the sub-Finsler isoperimetric problem in a restricted class of sets.

AN ISOPERIMETRIC RESULT FOR THE GAUSSIAN MEASURE AND UNCONDITIONAL SETS

2001 ◽  
Vol 33 (4) ◽  
pp. 408-416 ◽  
Author(s):  
F. BARTHE
Keyword(s):  
Gaussian Measure ◽  
Isoperimetric Problem ◽  
Boundary Measure

The paper studies an isoperimetric problem for the Gaussian measure and coordinatewise symmetric sets. The notion of boundary measure corresponding to the uniform enlargement is considered, and it is proved that symmetric strips or their complements have minimal boundary measure.

The isoperimetric problem

Resonance ◽  
1997 ◽  
Vol 2 (9) ◽  
pp. 65-68 ◽  
Author(s):  
Alladi Sitaram
Keyword(s):  
Isoperimetric Problem
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