Convergence of phase-field approximations to the Gibbs–Thomson law

We prove the convergence of phase-field approximations of the Gibbs–Thomson law. This establishes a relation between the first variation of the Van der Waals–Cahn–Hilliard energy and the first variation of the area functional. We allow for folding of diffuse interfaces in the limit and the occurrence of higher-multiplicities of the limit energy measures. We show that the multiplicity does not affect the Gibbs–Thomson law and that the mean curvature vanishes where diffuse interfaces have collided. We apply our results to prove the convergence of stationary points of the Cahn–Hilliard equation to constant mean curvature surfaces and the convergence of stationary points of an energy functional that was proposed by Ohta–Kawasaki as a model for micro-phase separation in block-copolymers.     

Synthesis and characterization of a series of diverse poly(2‐oxazoline)s

This article describes the synthesis and characterization of a variety of new poly(2‐oxazoline)s. With regard to functional polymers, 2‐oxazolines represent an interesting class of monomers because of the easy variation of the substituent in 2‐position. Starting from the corresponding nitriles, different 2‐oxazolines were obtained containing a diverse set of 2‐substituents, including thioether bonds ( M11 ), trifluoromethyl groups ( M8 , M10 ), and alkyl‐ or aryl groups ( M1 – M7 ). The subsequent polymerization of the majority of these monomers proceeded in a living manner, which was demonstrated by linear first‐order kinetics, a linear increase of molar mass with conversion, and relatively narrow molar mass distributions. In addition, selected thermal and surface properties of the polymers were studied utilizing DSC and contact‐angle measurements to determine the effects of different 2‐substituents on the macroscopic properties. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 3829–3838, 2009     

Irreducibility and Mixed Boundary Conditions

In this paper we consider positive semigroups on L^p(Ω) generated by elliptic operators A subject to mixed Dirichlet-Neumann boundary conditions on non-smooth domains Ω. We show in particular that these semigroups as well as those generated by multiplicative perturbations bA of A are irreducible, provided b ∈ L^∞(Ω) is real and satisfies b ≥ δ for some δ > 0. In memoriam Helmut H. Schaefer