LAYER SOLUTIONS FOR A ONE-DIMENSIONAL NONLOCAL MODEL OF GINZBURG LANDAU TYPE

被引：6
作者：
Chen, K. -S. [1 ]
Muratov, C. B. [2 ]
Yan, X. [1 ]
机构：
[1] Univ Connecticut, Dept Math, Storrs, CT 06269 USA
[2] New Jersey Inst Technol, Dept Math Sci, Newark, NJ 07102 USA
来源：
MATHEMATICAL MODELLING OF NATURAL PHENOMENA | 2017年 / 12卷 / 06期
基金：
美国国家科学基金会;
关键词：
Non-local Allen Cahn equation; layer solutions; De Giorgi conjecture; FRACTIONAL LAPLACIANS; NONLINEAR EQUATIONS; EXTENSION PROBLEM; 1-D SYMMETRY; DE-GIORGI; CONJECTURE; REGULARITY;
D O I：
10.1051/mmnp/2017068
中图分类号：
Q [生物科学];
学科分类号：
07 ; 0710 ; 09 ;
摘要：
We study a nonlocal model of Ginzburg Landau type that gives rise to an equation involving a mixture of the Laplacian and half-Laplacian. Our focus is on one-dimensional transition layer profiles that connect the two distinct homogeneous phases. We first introduce a renormalized one-dimensional energy that is free from a logarithmic divergence due to the failure of the Gagliardo norm to be finite on smooth profiles that asymptote to different limits at infinity. We then prove existence, uniqueness, monotonicity and regularity of minimizers in a suitable class. Lastly, we consider the singular limit in which the coefficient in front of the Laplacian vanishes and prove convergence of the obtained minimizer to the solutions of the fractional Allen Cahn equation.
引用
收藏
页码：68 / 90
页数：23
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