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 $s \in (0,1)$. 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 $C^{0,\alpha^*}$ regularity of the density, where the exponent $\alpha^*$ is explicit and is given by \begin{equation*} \alpha^* = \begin{cases} s & \text{for $s \in (0,1/2]$}\\ 2s-1 &\text{for $s \in (1/2,1]$}.\end{cases} \end{equation*} Under some additional assumptions, we then show that solutions are $C^{0,s}$. These results are optimal in the class of H\"older continuous functions. Thus, we find a complete correspondence with known results in case of the standard Laplacian.