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

Abstract

We are interested in the study of local and global minimizers for an energy functional of the type $$ \frac{1}{4} \iint_{\mathbb{R}^{2 N} \setminus \left( \mathbb{R}^N \setminus \Omega \right)^2} |u(x) - u(y)|^2 K(x - y) \, dx dy + \int_{\Omega} W(u(x)) \, dx, $$ 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 $K(z) = |z|^{- N - 2 s}$, 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 $$ \mbox{p.v.} \int_{\mathbb{R}^N} \left( u(x) - u(y) \right) K(x - y) \, dy + W'(u(x)) = 0 \quad \mbox{for any } x \in \mathbb{R}^N, $$ 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 $K(z) = |z|^{- N - 2 s}$. In our work, we consider instead general kernels which may be possibly non-homogeneous and truncated at infinity.