We study minimizers of Cahn-Hilliard energies under strong anchoring (Dirichlet) boundary conditions using a Nehari manifold approach with spectral analysis of the Dirichlet Laplacian.

For the de Gennes energy with quartic potential, we reveal bifurcation phenomena governed by the boundary value and transition layer thickness parameter. When the boundary value equals the phase average, and the parameter exceeds a critical threshold, the minimizer is unique and homogeneous; below this threshold, two symmetric minimizers emerge. Deviating boundary values restore uniqueness with asymmetric minimizers. We derive rigorous bounds for these solutions.

We extend this framework to the Flory-Huggins logarithmic potential, which models polymer blends and presents singular behavior at boundaries. Our analysis, supported by numerical simulations, reveals temperature-mediated bifurcations and demonstrates how the Nehari manifold technique provides a unified treatment of both functionals under strong anchoring conditions.