Coexistence and segregation for strongly competing species in special domains

Abstract. We deal with strongly competing multispecies systems of Lotka-Vol– terra type with homogeneous Dirichlet boundary conditions. We show that the shape of the spatial domain contributes to the occurrence of pattern formation (coexistence) and prevents extinction of the species as the competition grows in-definitely. As a result we prove the existence of complete solutions for a remarkable system of variational inequalities involved in segregation phenomena and optimal partition problems. 1