A symmetry result in ℝ² for global minimizers of a general type of nonlocal energy

2017/08/16 by Bucur, Claudia
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1708.04924

Abstract

In this paper, we are interested in a general type of nonlocal energy, defined on a ball B_R⊂ \mathbb Rⁿ for some R>0 as \mathcal E (u, B_R)= \iint_{\mathbb R²ⁿ∖ (\mathcal C BR)²} F( u(x)-u(y),x-y) dx dy+∫_BR W(u) dx. We prove that in \mathbb R², under suitable assumptions on the functions F and W, bounded continuous global energy minimizers are one-dimensional. This proves a De Giorgi conjecture for minimizers in dimension two, for a general type of nonlocal energy.

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