Abstract. Common models for two-phase lipid bilayer membranes are based on an energy that consists of an elastic term for each lipid phase and a line energy at interfaces. Although such an energy controls only the length of interfaces, the membrane surface is usually assumed to be at least C 1 across phase boundaries. We consider the spontaneous curvature model for closed rotationally symmetric two-phase membranes without excluding tangent discontinuities at interfaces a priorily. We introduce a family of energies for smooth surfaces and phase fields for the lipid phases and derive a sharp interface limit that coincides with the Γ-limit on all reasonable membranes and extends the classical model by assigning a bending energy also to tangent discontinuities. The theoretical result is illustrated by numerical examples.

We consider a diffuse interface approximation for the lipid phases of rotationally symmetric two-phase bilayer membranes and rigorously derive its Γ-limit. In particular, we prove that limit vesicles are C 1 across interfaces, which justifies a regularity assumption that is widely made in formal asymptotic and numerical studies. Moreover, a limit membrane may consist of several topological spheres, which are connected at the axis of revolution and resemble complete buds of the vesicle. 6,12,13,15,22]. Here H and K are the mean curvature and the Gauss curvature of the membrane surface M , and µ is its area measure. The bending rigidities k ± > 0 and the Gauss rigidities k ± G are elastic material parameters, and H ± s -the so-called spontaneous curvatures -are supposed to reflect an asymmetry in the membrane. In the simplest case, the rigidities and spontaneous curvatures are constant within each lipid phase but different between the two phases. The length of the phase interfaces ∂M + = ∂M − is denoted by H 1 (∂M + ), and σ is a line tension parameter. among all closed surfacesIn [15] the Euler-Lagrange equations of (1.1) for axially symmetric two-phase membranes with exactly one interface are studied. The authors mention the possibility of different smoothness conditions at the interface, their analysis, however, is done for smooth membranes, which are C 1 across the interface, only. Phase field models for the lipid phases and also the membrane surface are introduced in [10,11,28,17] and studied numerically; convergence to the sharp interface limit is obtained by asymptotic expansion and under additional smoothness assumptions and topological restrictions.In this paper we are interested in the convergence of a diffuse interface approximation for the lipid phases in a rotationally symmetric setting without imposing smoothness or the topological structure of limit vesicles in advance. More precisely, for a closed surface arXiv:1603.05231v1 [math.AP]

We study the evolution of diblock copolymer melts in which one component has small volume fraction. In this case one observes phase morphologies which consist of small spheres of the minority component embedded in the other component. Based on the Ohta-Kawasaki free energy one can set up an evolution equation which has the interpretation of a gradient flow. We restrict this gradient flow to morphologies in which the minority phase consists of spheres and derive monopole approximations for different parameter regimes. We use these approximations for simulations of large particle systems.

We study the motion of phase interfaces in a diffusive lattice equation with bistable nonlinearity and derive a free boundary problem with hysteresis to describe the macroscopic evolution in the parabolic scaling limit.The first part of the paper deals with general bistable nonlinearities and is restricted to numerical experiments and heuristic arguments. We discuss the formation of macroscopic data and present numerical evidence for pinning, depinning, and annihilation of interfaces. Afterwards we identify a generalized Stefan condition along with a hysteretic flow rule that characterize the dynamics of both standing and moving interfaces.In the second part, we rigorously justify the limit dynamics for single-interface data and a special piecewise affine nonlinearity. We prove persistence of such data, derive upper bounds for the macroscopic interface speed, and show that the macroscopic limit can indeed be described by the free boundary problem. The fundamental ingredient to our proofs is a representation formula that links the solutions of the nonlinear lattice to the discrete heat kernel and enables us to derive macroscopic compactness results in the space of continuous functions. Keywords:forward-backward diffusion in lattices, coarse graining for gradient flows, hysteretic models for phase transitions, pinning and depinning of interfaces, regularization of ill-posed parabolic PDEs MSC (2010):

We consider particles on a one-dimensional lattice whose evolution is governed by nearest-neighbor interactions where particles that have reached size zero are removed from the system. Concentrating on configurations with infinitely many particles, we prove existence of solutions under a reasonable density assumption on the initial data and show that the vanishing of particles and the localized interactions can lead to non-uniqueness. Moreover, we provide a rigorous upper coarsening estimate and discuss generic statistical properties as well as some non-generic behavior of the evolution by means of heuristic arguments and numerical observations.