SIAM Journal on Applied Mathematics, Volume 80, Issue 5, Page 2299-2317, January 2020.
We study analytically and numerically the minimizers for the Cahn--Hilliard energy functional with a symmetric quartic double-well potential and under a strong anchoring condition (i.e., the Dirichlet condition) on the boundary of an underlying bounded domain. We show a bifurcation phenomenon determined by the boundary value and a parameter that describes the thickness of a transition layer separating two phases of an underlying system of binary mixtures. For the case that the boundary value is exactly the average of the two pure phases, if the bifurcation parameter is larger than or equal to a critical value, then the minimizer is unique and is exactly the homogeneous state. Otherwise, there are exactly two symmetric minimizers. The critical bifurcation value is inversely proportional to the first eigenvalue of the negative Laplace operator with the zero Dirichlet boundary condition. For a boundary value that is larger (or smaller) than that of the average of the two pure phases, the symmetry is broken and there is only one minimizer. We also obtain the bounds and morphological properties of the minimizers under additional assumptions on the domain. Our analysis utilizes the notion of the Nehari manifold and connects it to the eigenvalue problem for the negative Laplacian with the homogeneous boundary condition. We numerically minimize the functional $E$ by solving the gradient-flow equation of $E$, i.e., the Allen--Cahn equation, with the designated boundary conditions, and with random initial values. We present our numerical simulations and discuss them in the context of our analytical results.

中文翻译：

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强锚定条件下Cahn-Hilliard能量函数的最小化器

SIAM应用数学杂志，第80卷，第5期，第2299-2317页，2020年1月。
我们通过分析和数值研究具有对称四次双势势且在基本有界域边界上的强锚定条件（即Dirichlet条件）下的Cahn-Hilliard能量函数的极小值。我们显示了由边界值和描述过渡层厚度的参数确定的分叉现象，该过渡层分隔了基础二元混合物系统的两个相。对于边界值恰好是两个纯相的平均值的情况，如果分叉参数大于或等于临界值，则最小化器是唯一的，并且恰好是均匀状态。否则，恰好有两个对称的最小化器。临界分叉值与Dirichlet边界条件为零的负Laplace算子的第一特征值成反比。对于大于（或小于）两个纯相平均值的边界值，对称性将被破坏，并且只有一个最小化器。在该域上的其他假设下，我们还获得了极小值的边界和形态学特性。我们的分析利用了Nehari流形的概念，并将其与具有齐次边界条件的负Laplacian的特征值问题联系起来。我们通过求解带有指定边界条件和随机初始值的$ E $梯度流方程（即Allen-Cahn方程）在数值上最小化函数$ E $。