  In this paper we study an optimal shape design problem for the first
eigenvalue of the fractional $p-$laplacian with mixed boundary conditions. The
optimization variable is the set where the Dirichlet condition is imposed (that
is restricted to have measure equal than a prescribed quantity, $\alpha$). We
show existence of an optimal design and analyze the asymptotic behavior when
the fractional parameter $s\uparrow 1$ obtaining asymptotic bounds that are
independent of $\alpha$.
