Gamma Convergence of Partially Segregated Elliptic Systems

Abstract

We study partially segregated elliptic systems through the use of penalized energy functionals. These systems arise from the minimization of Gross-Pitaevskii-type energies that capture the behavior of multi-component ultracold gas mixtures and other systems involving multiple interacting fluid or gas species. In the case when the domain is planar, i.e., in $\mathbb{R}^2$, our main result is the Gamma convergence of penalized energy to the constrained Dirichlet energy with strict segregation. The proof combines lower semicontinuity arguments with a recovery sequence construction based on geometric decompositions near interfaces and triple junctions. This establishes a rigorous variational link between the penalized and constrained formulations.