In this paper, we are interested in a general type of nonlocal energy, defined on a ball BR⊂ Rn for some R> 0 as E(u,BR)=∬R2n\(CBR)2F(u(x)-u(y),x-y)dxdy+∫BRW(u)dx.We prove that in 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.

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

Bucur C.
2020-01-01

Abstract

In this paper, we are interested in a general type of nonlocal energy, defined on a ball BR⊂ Rn for some R> 0 as E(u,BR)=∬R2n\(CBR)2F(u(x)-u(y),x-y)dxdy+∫BRW(u)dx.We prove that in 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.
2020
Nonlocal energy; one dimensional solutions; Allen-Cahn; De Giorgi conjecture
Bucur, C.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2105129
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