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 / C. Bucur. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - 59:2(2020), pp. 52.1-52.26. [10.1007/s00526-020-1698-6]
A symmetry result in R2 for global minimizers of a general type of nonlocal energy
C. Bucur
Primo
2020
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.
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