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 $$\frac{1}{4}{\iint }_{{\mathbb{R}}^{2N}\setminus {\left({\mathbb{R}}^N\setminus \Omega \right)}^2}|u(x)-u(y){|}^{2}K(x-y)dxdy+{\int }_{\Omega }W(u(x))dx,$$ 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 $K(z)=|z{|}^{-N-2s}$, 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 $$ \mbox{p.v.} \int_{\mathbb{R}^N} \left( u(x) - u(y) \right) K(x - y) \, dy + W'(u(x)) = 0 \quad \mbox{for any } x \in \mathbb{R}^N, $$ which possess one-dimensional symmetry. The results presented here were proved in (Cabré and Solà-Morales, 2005), (Palatucci, Savin and Valdinoci, 2013) and (Cabré and Sire, 2015) for the model case $K(z)=|z{|}^{-N-2s}$. In our work, we consider instead general kernels which may be possibly non-homogeneous and truncated at infinity.

References in corpus (4)

• Nonlocal diffusion and applications
• Local behavior of fractional $p$-minimizers
• Nonlocal equations with measure data
• Plane-like minimizers for a non-local Ginzburg-Landau-type energy in a periodic medium

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