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 type14∬R2N∖(RN∖Ω)2|u(x)−u(y)|2K(x−y)dxdy+∫ΩW(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∈(0,1), related to the fractional Laplacian. We show the existence and uniqueness (up to translations) of one-dimensional minimizers in the full space RN and obtain sharp estimates for some quantities associated to it. In particular, we deduce the existence of solutions of the non-local Allen–Cahn equationp.v.∫RN(u(x)−u(y))K(x−y)dy+W′(u(x))=0for any x∈RN, which possess one-dimensional symmetry.The results presented here were proved in [9,10,36] 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.