Phase Transition for Potentials of High-Dimensional Wells with a Mass-Type Constraint

Abstract

Inspired by Lin-Pan-Wang (Comm. Pure Appl. Math., 65(6): 833-888, 2012), we continue to study the corresponding time-independent case of the Keller-Rubinstein-Sternberg problem. To be precise, we explore the asymptotic behavior of minimizers as $\epsilon \to 0$, for the functional $${\mathbf{E}}_{\epsilon }(u)={\int }_{\hat{I}}\text{c◯}\left(|\nabla u{|}^2+\frac{1}{{\epsilon }^2}F(u)\right)dx$$under a mass-type constraint ${\int }_{\hat{I}}\text{c◯}\ddot{I}(u)dx=m$, where $\ddot{I}:{\mathbb{R}}^k\to \mathbb{R}\in Lip({\mathbb{R}}^k)$ is specialized as a density function with $m$ representing a fixed total mass. The potential function $F$ vanishes on two disjoint, compact, connected, smooth Riemannian submanifolds $N^{\pm }\subset {\mathbb{R}}^k$. We analyze the expansion of ${\mathbf{E}}_{\epsilon }(u_{\epsilon })$ for various density functions $\ddot{I}$, 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 different geometric assumptions and characterize the convergence $u_{{\epsilon }_i}\to v$ in the $L^1$ sense.