Minimizing Sequences Selected via Singular Perturbations, and their Pattern Formation

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We study the Cahn-Hilliard energy E ɛ(u) over the unit square under the constraint of a constant mass m with (ɛ > 0) and without ɛ= 0) interfacial energy. Minimizers of E 0(u) have no preferred pattern and we select patterns via sequences of conditionally critical points of E ɛ(u) converging to minimizers as ɛ tends to zero. Those critical points are not minimizers if the singular limit has no minimal interface. We obtain them by a global bifurcation analysis of the Euler-Lagrange equations for E ɛ(u) where the mass m is the bifurcation parameter. We make use of the symmetry of the unit square, and the elliptic maximum principle, in turn, implies that the location of maxima and minima is fixed for all solutions on global branches. This property is used to guarantee the existence of a singular limit and to verify the Weierstrass-Erdmann corner condition which proves its minimizing property.