Connectivity of Phase Boundaries in Strictly Convex Domains
Connectivity of Phase Boundaries in Strictly Convex Domains
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DOI：
10.1007/s002050050081
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发表时间：
1998-04
影响因子：
2.5
通讯作者：
P. Sternberg;K. Zumbrun
中科院分区：
文献类型：
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作者：
P. Sternberg;K. Zumbrun
我们考虑相变模型中的平衡点，它对应于约束变分问题的稳定临界点，其中W是双阱势，$\Omega\subset\R^n$是严格凸域。对于ε小，这与将Ω划分为两个固定体积的子域的问题密切相关，其中子域边界对应于相之间的过渡边界。出于这个几何问题，我们表明，在一个严格的凸域，稳定的临界点的原始变分问题有一个连接，薄过渡层分离的两个阶段。这与[GM]中的工作有关，[GM]中处理了特殊的几何形状，如圆柱形区域，并且类似于[CHo]中的结果，该结果表明在凸区域中，相应的无约束问题的稳定临界点是常数。连通性的证明使用了几何测度理论的工具，包括共同面积公式和流形上的等周不等式。过渡层的薄度来自建立纯相溶液的空间衰减的单独计算。
We consider equilibria arising in a model for phase transitions which correspond to stable critical points of the constrained variational problemHereWis a double‐well potential and $\Omega\subset\R^n$ is a strictly convex domain. Forεsmall, this is closely related to the problem of partitioningΩinto two subdomains of fixed volume, where the subdomain boundaries correspond to the transitional boundary between phases. Motivated by this geometry problem, we show that in a strictly convex domain, stable critical points of the original variational problem have a connected, thin transition layer separating the two phases. This relates to work in [GM] where special geometries such as cylindrical domains were treated, and is analogous to the results in [CHo] which show that in a convex domain, stable critical points of the correspondingunconstrainedproblem are constant. The proof of connectivity employs tools from geometric measure theory including the co‐area formula and the isoperimetric inequality on manifolds. The thinness of the transition layer follows from a separate calculation establishing spatial decay of solutions to the pure phases.