Gamma Convergence and Smooth Exact-segregation Approximation for Partially Segregated Systems

Abstract

We study three-component Dirichlet energies subject to the partial segregation constraint \(u_1u_2u_3=0\). Our main result is a smooth approximation theorem on bounded Lipschitz domains in arbitrary dimension. For compatible nonnegative Lipschitz boundary data, every admissible \(H^1\) configuration can be approximated strongly in \(H^1\) by globally Lipschitz maps that are smooth in the interior and preserve both the prescribed boundary trace and the segregation constraint exactly. The construction combines a Lipschitz retraction, variable-radius mollification, and smooth truncation preserving existing zeros, without assumptions on interfaces or junctions. Under the weaker assumption of compatible nonnegative \(H^{1/2}\) boundary data, we establish existence of minimizers and \(Γ\)-convergence in \(L^2\) for the energies with product penalization \(\varepsilon^{-1}\int_Ω(u_1u_2u_3)^2\,dx\). We also identify the exact \(L^2\) distance to the constrained class and construct segregated corrections with controlled Dirichlet energy. The resulting correction estimate has order \(\varepsilon^{1/6}\) for bounded-energy configurations; this exponent is sharp in general and improves to \(o(\varepsilon^{1/6})\) for almost-minimizers.

arXiv comment25 pages; Keywords: Gamma convergence, partial segregation, smooth approximation, penalty methods, free boundary