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 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 equation 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.