One-dimensional solutions of non-local Allen-Cahn-type equations with rough kernels

We are interested in the study of local and global minimizers for an energy functional of the type $$\frac{1}{4}{\iint }_{{\mathbb{R}}^{2N}\setminus {({\mathbb{R}}^N\setminus \mathrm{\Omega })}^2}\mathrm{\mid }u(x)-u(y){\mathrm{\mid }}^{2}K(x-y)dxdy+{\int }_{\mathrm{\Omega }}W(u(x))dx,$$ where $W$ is a smooth, even double-well potential and $K$ is a non-negative symmetric kernel in a general class, which contains as a particular case the choice $K(z)=\mathrm{\mid }z{\mathrm{\mid }}^{-N-2s}$, with $s \in (0, 1)$, related to the fractional Laplacian. We show the existence and uniqueness (up to translations) of one-dimensional minimizers in the full space ${\mathbb{R}}^N$ and obtain sharp estimates for some quantities associated to it. In particular, we deduce the existence of solutions of the non-local Allen-Cahn equation $$p\mathrm{.}v\mathrm{.}{\int }_{{\mathbb{R}}^N}(u(x)-u(y))K(x-y)dy+W^{\mathrm{\prime }}(u(x))=0foranyx\in {\mathbb{R}}^N,$$ which possess one-dimensional symmetry. The results presented here were proved in (Cabr'e and Sol`a-Morales, 2005), (Palatucci, Savin and Valdinoci, 2013) and (Cabr'e and Sire, 2015) for the model case $K(z)=\mathrm{\mid }z{\mathrm{\mid }}^{-N-2s}$. In our work, we consider instead general kernels which may be possibly non-homogeneous and truncated at infinity.