*Published Paper*

**Inserted:** 1 nov 2023

**Last Updated:** 1 nov 2023

**Journal:** Nonlinear Analysis

**Year:** 2019

**Doi:** 10.1016/j.na.2019.05.013

**Abstract:**

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