We present a formal asymptotic analysis which suggests a model for three-phase boundary motion as a singular limit of a vector-valued Ginzburg-Landau equation. We prove short-time existence and uniqueness of solutions for this model, that is, for a system of three-phase boundaries undergoing curvature motion with assigned angle conditions at the meeting point. Such models pertain to grain boundary motion in alloys. The method we use, based on linearization about the initial conditions, applies to a wide class of parabolic systems. We illustrate this further by its application to an eutectic solidification problem.

In this paper we study a recently proposed model for the growth of a nonnecrotic, vascularized tumor. The model is in the form of a free-boundary problem whereby the tumor grows (or shrinks) due to cell proliferation or death according to the level of a diffusing nutrient concentration. The tumor is assumed to be spherically symmetric, and its boundary is an unknown function r = s(t). We concentrate on the case where at the boundary of the tumor the birth rate of cells exceeds their death rate, a necessary condition for the existence of a unique stationary solution with radius r = R0 (which depends on the various parameters of the problem). Denoting by c the quotient of the diffusion time scale to the tumor doubling time scale, so that c is small, we rigorously prove that (i) lim inf s(t) > 0, i.e. once engendered, tumors persist in time. t-->infinity Indeed, we further show that (ii) If c is sufficiently small then s(t)-->R0 exponentially fast as t-->infinity, i.e. the steady state solution is globally asymptotically stable. Further, (iii) If c is not "sufficiently small" but is smaller than some constant gamma determined explicitly by the parameters of the problem, then t-->infinity lim sup s(t) < infinity; if however c is "somewhat" larger than gamma then generally s(t) does not remain bounded and, in fact, s(t)-->infinity exponentially fast as t-->infinity.

We present a new algorithm for the numerical solution of problems of electromagnetic or acoustic scattering by large, convex obstacles. This algorithm combines the use of an ansatz for the unknown density in a boundary-integral formulation of the scattering problem with an extension of the ideas of the method of stationary phase. We include numerical results illustrating the high-order convergence of our algorithm as well as its asymptotically bounded computational cost as the frequency increases.

This paper outlines the theoretical background of a new approach towards an accurate and well-conditioned perturbative calculation of Dirichlet{Neumann operators (DNOs) on domains that are perturbations of simple geometries. Previous work on the analyticity of DNOs has produced formulae that, as we have found, are very ill-conditioned. We show how a simple change of variables can lead to recursions that satisfy analyticity estimates without relying on subtle cancellation properties at the heart of previous formulae.