Introduction

Abstract

The Lotka–Volterra competition model from biological mathematics, which describes the dynamics of several species competing with each other, and the coupled Gross–Pitaevskii equations arising from Bose–Einstein condensates in theoretical physics, involve a class of systems of singularly perturbed elliptic or parabolic partial differential equations. The common feature in these problems is that there is a parameter κ. When κ is finite, we can observe partial overlap between different species (or condensates), and when κ goes to infinity (strong competition or large interaction), different species or condensates tend to be disjoint. This is known as the phase separation phenomena. This proposes a new class of free boundary problems, which is also related to the harmonic map into a singular space with non-positive curvature. In recent years, these problems have attracted a lot of interests. Many mathematicians, including L. Caffarelli, E.N. Dancer, F.H. Lin and S. Terracini have obtained a lot of deep results in this direction. In this chapter, we first introduce these problems. Then we recall some known results. In the last section, we list the main results in this thesis.