## The Mathematical Modeling and Analysis of Complex Structures in Materials Science

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In materials science, there are many complex structures due to different components of the system. The Cahn--Hilliard (CH) equation is a common model to describe phase-separation processes of a mixture of two components. The Functionalized Cahn--Hilliard (FCH) equation has been proposed as a model for the interfacial energy of phase-separated mixtures of amphiphilic molecules. We want to study theoretical properties of the CH and FCH equations, such as the existence, uniqueness, and corresponding geometric evolution governed by weak solutions, so that we can get a better understanding of complex structures.First, we study the effect of the strong anchoring condition on the minimizers of the CH energy functional with a symmetric quartic double-well potential. 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.Second, we study the well-posedness of the CH equation coupled with the homogeneous strong anchoring condition and no-flux boundary condition. With a specific quartic form of the double-well potential, we prove the existence and uniqueness of the weak solution to this model by interpreting the problem as a gradient flow of the CH free energy. Third, we study the existence of a nonnegative weak solution for the FCH equation subject to a cutoff degenerate diffusion mobility M(u). Assuming the initial data is positive, we construct a nonnegative weak solution as the limit of solutions corresponding to non-degenerate mobilities and verify that it satisfies an energy dissipation inequality. Lastly, we want to explore the geometric evolution of bilayers under the degenerate FCH equation. With the same cutoff degenerate mobility as in the third result, using multi-scale analysis, we derive a sharp interface limit for the dynamics of bilayer structures of the degenerate FCH equation.