  We deal with strongly competing multispecies systems of Lotka-Volterra type
with homogeneous Dirichlet boundary conditions. For a class of nonconvex
domains composed by balls connected with thin corridors, we show the occurrence
of pattern formation (coexistence and spatial segregation of all the species),
as the competition grows indefinitely. As a result we prove the existence and
uniqueness of solutions for a remarkable system of differential inequalities
involved in segregation phenomena and optimal partition problems.
