Layer solutions for a one-dimensional nonlocal model of Ginzburg-Landau type

Document Type

Article

Publication Date

1-1-2017

Abstract

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.

Identifier

85040370817 (Scopus)

Publication Title

Mathematical Modelling of Natural Phenomena

External Full Text Location

https://doi.org/10.1051/mmnp/2017068

e-ISSN

17606101

ISSN

09735348

First Page

68

Last Page

90

Issue

6

Volume

12

Grant

1614948

Fund Ref

National Science Foundation

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