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∈(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 C0,αjavax.xml.bind.JAXBElement@41b1f147 regularity of the density, where the exponent α∗ is explicit and is given by α∗=sfors∈(0,1∕2]2s−1fors∈(1∕2,1).Under some additional assumptions, we then show that solutions are C0,s. These results are optimal in the class of Hölder continuous functions. Thus, we find a complete correspondence with known results in case of the standard Laplacian.