Well-posedness and a sign-graph limit for an active Cahn--Hilliard equation on Riemannian manifolds
We establish a weak solution theory and a singular constitutive limit for a non variational Cahn Hilliard evolution on compact Riemannian manifolds, possibly with boundary. The model combines the classical double well chemical potential with an active correction concentrated in diffuse transition layers and sensitive to the sign of the Laplace Beltrami operator, together with a monotone anchoring mechanism toward a prescribed reference state. The active correction breaks the passive gradient flow structure and introduces a nonlinear dependence on second derivatives not identified by the natural weak compactness estimates. For square integrable initial data and sufficiently integrable reference data, we construct weak solutions in arbitrary dimension. A Galerkin level one sided comparison argument, using monotonicity of the classifier and anchoring law together with biharmonic coercivity, yields strong convergence of the approximate Laplacians and hence strong second order compactness. This identifies the nonlinear active term in an ordinary weak formulation, without a second derivative defect. In two dimensions we prove uniqueness and continuous dependence. The estimates depend only on the bound and monotonicity of the classifier, not on its slope, and are therefore uniform for increasingly steep arctangent classifiers. Their singular limit is governed by the maximal monotone sign graph acting on the Laplace Beltrami operator. In arbitrary dimension we obtain subsequential strong second order convergence to a weak solution of the resulting differential inclusion. In two dimensions the limiting state is unique, so the entire steep classifier family converges; uniqueness of the constitutive multiplier on the zero Laplacian set is not asserted. This is a constitutive steepness limit at fixed diffuse interface thickness, rather than a sharp interface limit.