Heteroclinic connections for fractional Allen-Cahn equations with degenerate potentials

TL;DR

This paper studies the existence, uniqueness, and behavior of heteroclinic connections in fractional Allen-Cahn equations with degenerate potentials, using non-local energy functionals involving fractional Laplacian-like kernels.

Contribution

It introduces a framework for analyzing heteroclinic connections in fractional Allen-Cahn equations with degenerate potentials and broad kernel classes.

Findings

Existence of minimizers for the non-local energy functionals.

Uniqueness and asymptotic behavior of these minimizers.

Characterization of heteroclinic connections in the fractional setting.

Abstract

We investigate existence, uniqueness and asymptotic behavior of minimizers of a family of non-local energy functionals of the type $$ \frac{1}{4}\iint_{\mathbb{R}^{2n}\setminus (\mathbb{R}^n \setminus \Omega)^2}|u(x)-u(y)|^2 K(x-y) \,dx dy + \int_\Omega W(u(x)) \,dx. $$ Here, $W$ is a possibly degenerate double well potential with a polynomial control on its second derivative near the wells. Also, ${K}$ belongs to a wide class of measurable kernels and is modeled on that of the fractional Laplacian.