| Issue |
Math. Model. Nat. Phenom.
Volume 12, Number 6, 2017
Special Issue - Nonlocal and delay equations
|
|
|---|---|---|
| Page(s) | 68 - 90 | |
| DOI | https://doi.org/10.1051/mmnp/2017068 | |
| Published online | 30 December 2017 | |
Layer solutions for a one-dimensional nonlocal model of Ginzburg–Landau type
1
Department of Mathematics, University of Connecticut,
Storrs,
CT
06269, USA
2
Department of Mathematical Sciences, New Jersey Institute of Technology,
Newark,
NJ
07102, USA
Received:
11
October
2017
Accepted:
11
October
2017
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.
Mathematics Subject Classification: 35R11 / 35J20 / 49J20
Key words: Non-local Allen–Cahn equation / layer solutions / De Giorgi conjecture
© EDP Sciences, 2017
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