Phase Transition for Potentials of High-Dimensional Wells with a Mass-Type Constraint
Xingyu Wang, Yaguang Wang
TL;DR
This work extends the scalar and low-dimensional phase-transition theory to vector-valued Wells vanishing on two manifolds $N^{\pm}$ under a mass-type constraint. By combining $BV$-based co-area techniques with the minimal-connection framework, the authors establish a precise leading-order energy law $\lim_{\varepsilon\to0} \varepsilon \mathbf{E}_{\varepsilon} = c_0^F I_{\Omega}(\sigma_m)$, where $c_0^F$ encodes the one-dimensional optimal transition and $I_{\Omega}$ is the domain’s isoperimetric profile minimized at $\sigma_m$. Under additional geometric assumptions (e.g., $n\le7$ and a nondegenerate isoperimetric profile near $\sigma_m$), a refined two-term expansion $\mathbf{E}_{\varepsilon}=\dfrac{c_0^F I_{\Omega}(\sigma_m)}{\varepsilon}+O(1)$ is obtained, and minimizers $u_{\varepsilon}$ converge in $L^1$ to a sharp partition $v$ with $v\in H^1(\Omega^{\pm},N^{\pm})$. The paper also treats Type II mass constraints, where a Pólya–Szegő reduction yields one-dimensional reductions and analogous energy expansions, highlighting the role of curvature and symmetry in the higher-order terms. These results rigorously justify isoperimetric-type interface limits and quantify how domain geometry and mass constraints shape phase separation in high-dimensional vector-valued wells.
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 $\varepsilon\to0$, for the functional $$\mathbf{E}_\varepsilon(u)= \int_Ω\left(|\nabla u|^{2}+\frac{1}{\varepsilon^{2}} F(u)\right) d x$$under a mass-type constraint $\int_Ωρ(u)\, dx=m$, where $ρ:\mathbb{R}^k \to \mathbb{R}\in Lip(\mathbb{R}^k)$ 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 $\mathbf{E}_\varepsilon(u_\varepsilon)$ for various density functions $ρ$, 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_{\varepsilon_i}\to v $ in the ${L}^1$ sense.