Regularity results for segregated configurations involving fractional Laplacian

We study the regularity of segregated profiles arising from competition-diffusion models, where the diffusion process is of nonlocal type and is driven by the fractional Laplacian of power s is an element of (0, 1). Among others, our results apply to the regularity of the densities of an optimal partition problem involving the eigenvalues of the fractional Laplacian. More precisely, we show C-0,C-alpha* regularity of the density, where the exponent alpha* is explicit and is given by alpha* = {s for s is an element of (0, 1/2] 2s - 1 for s is an element of (1/2, 1). Under some additional assumptions, we then show that solutions are C-0,C-s. These results are optimal in the class of Holder continuous functions. Thus, we find a complete correspondence with known results in case of the standard Laplacian. (C) 2019 Elsevier Ltd. All rights reserved.