Phase transition for potentials of high-dimensional wells with a mass-type constraint

Inspired by Lin, Pan and Wang (Comm. Pure Appl. Math., 65(2012), pp. 833-888), we study the corresponding time-independent case of the Keller-Rubinstein-Sternberg problem. To be precise, we explore the asymptotic behavior of minimizers u_ε, as ε→0, for the functional [[EQUATION]] under a general ``total mass" constraint ∫Ωρ(u)\, dx=m, with ρ∈ Lip(Rᵏ,R) being a given density function for a fixed total mass m. The potential function F vanishes on two disjoint, compact, connected, smooth Riemannian submanifolds N±ᵏ. We analyze the expansion of E_ε(u_ε), identifying the leading-order term in the asymptotic expansion, which depends on the geometry of the domain and the energy of minimal connecting orbits between N⁺ and N⁻. Furthermore, we estimate the higher-order term under special geometric assumptions and characterize the convergence of subsequences uεᵢ→ u in the L¹ sense.