Axisymmetric critical points of a nonlocal isoperimetric problem on the two-sphere

Abstract

Abstract. On the two dimensional sphere, we consider axisymmetric critical points of an isoperimetric problem perturbed by a long-range interaction term. When the parameter controlling the nonlocal term is sufficiently large, we prove the existence of a local minimizer with arbitrary many interfaces in the axisymmetric class of admissible functions. These local minimizers in this restricted class are shown to be critical points in the broader sense (i.e., with respect to all perturbations). We then explore the rigidity, due to curvature effects, in the criticality condition via several quantitative results regarding the axisymmetric critical points.

Singular perturbation of an elastic energy with a singular weight

“…Of particular interest is the connection of the functionals (1.1) with the Ohta-Kawasaki theory of diblock copolymers (cf. [3,6,13]) on the surface of the unit two-sphere [4,16]. Pattern formation of ordered structures on curved surfaces arises in systems ranging from biology to materials science.…”

“…There is extensive literature on the mathematical analysis of the Ohta-Kawasaki model on flat domains, such as the flattori, general bounded domains, and the unbounded Euclidean space (see [5] for a review). The rigorous mathematical analysis of the Ohta-Kawasaki model on curved spaces is rather rare [4,16].…”

“…by a change of variables t = cos φ (see [4,Section 2] for details). After another suitable change of variables, minimization of E ε with α = 1/2 and β = 1 is equivalent to minimization of energies E ε over H 1 -functions which satisfy |u | = 1 and u(−1) = u(1) = 0.…”

Sharp interface limit of an energy modelling nanoparticle-polymer blends

“…There is an extensive literature on the mathematical analysis of phase separation of block copolymers via the Ohta-Kawasaki model and its sharp interface limit leading to a nonlocal isoperimetric problem. From mathematical derivation of the model [8,12] to analysis on curved manifolds [15,56], the energy landscape of (1.3) with η = 0 and its Γ-limit as → 0 whether posed on the flat torus (i.e. with periodic boundary conditions), on a general domain with homogeneous Neumann data or on the whole Euclidean space has been rigorously investigated in various parameter regimes of m and γ (cf.…”