the dynamics of three phase triple junction and contact points we use the method of matched asymptotic expansions to study the sharp interface limit of the three phase system modeled by the cahn hilliard equations with the relaxation boundary condition the dynamic laws for the interfaces the triple junction and the contact points are derived at different time scales in particular we show at o t time scale the dynamic of the triple junction is determined by the balance of the chemical potential gradient along the three interfaces meeting at the triple junction at faster time scale o epsilon t the motion of the triple junction is controlled by the contact point motions and geometric constraints

bifurcation of binary systems with the onsager mobility the main objective of this article is to study the effect of the nonlinear onsager mobility to the phase separation of the binary system using rigorous bifurcation analysis in particular a nondimensional parameter k depending on the molar density u0 of the homogeneous state and the critical temperature is derived the sign of this parameter dictates the type of transition also the analysis indicates that the type of the transition the critical temperature tc and the strength of the deviation of the transition solutions from the homogeneous state are all independent of the choices of the onsager mobility

existence of solutions for a class of fourth order cross diffusion systems of gradient flow type abstract this article is concerned with the existence and the long time behavior of weak solutions to certain coupled systems of fourth order degenerate parabolic equations of gradient flow type the underlying metric is a wasserstein like transportation distance for vector valued functions with nonlinear mobilities in each component under the hypothesis of flat convexity of the driving free energy functional weak solutions are constructed by means of the variational minimizing movement scheme for metric gradient flows the essential regularity estimates are derived by variational methods

finite element approximation of the cahn hilliard equation with degenerate mobility we consider a fully practical finite element approximation of the cahn hilliard equation with degenerate mobility textstyle frac partial u partial t del b u del gamma lap u psi u where b cdot geq 0 is a diffusional mobility and psi cdot is a homogeneous free energy in addition to showing well posedness and stability bounds for our approximation we prove convergence in one space dimension furthermore an iterative scheme for solving the resulting nonlinear discrete system is analyzed we also discuss how our approximation has to be modified in order to be applicable to a logarithmic homogeneous free energy finally some numerical experiments are presented

singularities of linear systems and the waring problem the waring problem for homogeneous forms asks for additive decomposition of a form f into powers of linear forms a classical problem is to determine when such a decomposition is unique in this paper we answer this question when the degree of f is greater than the number of variables to do this we translate the algebraic statement into a geometric one concerning the singularities of linear systems of p with assigned singularities

a practically unconditionally gradient stable scheme for the n component cahn hilliard system we present a practically unconditionally gradient stable conservative nonlinear numerical scheme for the n component cahn hilliard system modeling the phase separation of an n component mixture the scheme is based on a nonlinear splitting method and is solved by an efficient and accurate nonlinear multigrid method and the scheme allows us to convert the n component cahn hilliard system into a system of n 1 binary cahn hilliard equations and significantly reduces the required computer memory and cpu time we observe that our numerical solutions are consistent with the linear stability analysis results we also demonstrate the efficiency of the proposed scheme with various numerical experiments

systems of cahn hilliard equations the phase separation of alloys with two or more components is studied with emphasis on more than two components particular attention is given to differences between multicomponent and binary alloys specific topics of the paper include equilibrium theory aspects of the dynamics and numerical simulations in the equilibrium theory it is found that there is an enriched equilibrium structure that allows for multiple coexisting phases and the presence of triple points in the solution dynamic results include the characterization of the spinodal region and of the concentration variations that lead to spinodal decomposition unlike the binary theory not all composition variations lead to separation and the compositions are not restricted to the convex hull of the equilibrium concentrations linear analysis is used to predict that a pseudo binary will initially result from spinodal decomposition numerical simulations of the dynamics for a ternary alloy verify this initially but more than two phases often

diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix a model for phase separation in multicomponent systems is studied in particular the possibility of a concentration dependence of the mobility matrix is taken into account this leads to a system of fourth order degenerate parabolic partial differential equations we derive these equations from balance laws show some properties of the model and prove a global existence result for the degenerate system

geometric motion for a degenerate allen cahn cahn hilliard system the partial wetting case abstract using formal asymptotics we demonstrate that in a low temperature coarsening limit a degenerate allen cahn cahn hilliard system yields a geometric problem in which small particles whose shape evolves according to surface diffusion move along a surface where the chemical potential is quasi static which itself moves by motion by mean curvature the degenerate allen cahn cahn hilliard system was developed in j w cahn a novick cohen evolution equations for phase separation and ordering in binary alloys j stat phys 76 1994 877 909 to describe simultaneous ordering and phase separation and within this context the particles which contain a minor disordered phase are embedded along grain boundaries which partition the system into two ordered phase variants the limiting problem though can also be viewed as a diffuse interface approximation for various problems in materials science in which surface diffusion and motion by mean curvature are coupled see for example j kanel a novick cohen a vilenkin a numerical study of grain boundary motion in bicrystals acta mat 53 2004 227 235 a novick cohen order disorder and phase separation modeling grain sintering in e inan k z markov eds proceedings of the ninth international symposium cmds9 istanbul turkey world scientific publishing co 1999 pp 56 68 the present analysis extends a previous study a novick cohen triple junction motion for an allen cahn cahn hilliard system physica d 137 2000 1 24 which focused on the complete wetting limit and on motions in the plane here we treat the partial wetting case and our analysis accommodates motion in three dimensions

multi component cahn hilliard system with different boundary conditions in complex domains abstract we propose an efficient phase field model for multi component cahn hilliard ch systems in complex domains the original multi component cahn hilliard system with a fixed phase is modified in order to make it suitable for complex domains in the cartesian grid along with contact angle or no mass flow boundary conditions on the complex boundaries the proposed method uses a practically unconditionally gradient stable nonlinear splitting numerical scheme further a nonlinear full approximation storage multigrid algorithm is used for solving semi implicit formulations of the multi component ch system incorporated with an adaptive mesh refinement technique the robustness of the proposed method is validated through various numerical simulations including multi phase separations via spinodal decomposition equilibrium contact angle problems and multi phase flows with a background velocity field in complex domains

a multi phase mullins sekerka system matched asymptotic expansions and an implicit time discretisation for the geometric evolution problem we propose a generalisation of the mullins sekerka problem to model phase separation in multi component systems the model includes equilibrium equations in bulk the gibbs thomson relation on the interfaces young s law at triple junctions together with a dynamic law of stefan type using formal asymptotic expansions we establish the relationship to a transition layer model known as the cahn hilliard system we introduce a notion of weak solutions for this sharp interface model based on integration by parts on manifolds together with measure theoretical tools through an implicit time discretisation we construct approximate solutions by stepwise minimisation under the assumption that there is no loss of area as the time step tends to zero we show the existence of a weak solution

nonnegative solutions of a fourth order nonlinear degenerate parabolic equation elena beretta michiel bertsch roberta dal passo communicated by m guatin o introduction in this paper we study nonnegative solutions of the fourth order equation 0 1 ut u uxxx x o where n is a real positive constant equation 0 1 is a nonlinear parabolic partial differential equation which is degenerate at points at which u vanishes it arises in several applications see the references in 2 4 5 6 in which u x t stands for a nonnegative quantity for example in the study of the dynamics of thin liquid films u denotes the thickness of the film or another relevant nonnegative quantity the case n 1 models the flow in a hele shaw cell while n 3 corresponds to viscous flow on a solid surface without slip driven by surface tension see also 6 7 8 for a related equation it is well known that fourth order equations do not satisfy a maximum principle in particular classical solutions of the linear equation ut u 0 may become negative in finite time bernis 8 friedman 2 have observed that if n is positive equation 0 1 has generalized solutions which remain nonnegative for all times in addition they show that the qualitative positivity of these solutions strongly depends on the value of n in more recent papers by kadanoff et al a combination of numerical calculations and matched asymptotics 4 and the analysis of travelling wave solutions 5 leads to results which suggest a very rich structure of the solutions of 0 1 it is the purpose of the present paper to clarify several aspects of the nonlinear equation 0 1 and its dependence on the parameter n our results lead to several conjectures and open problems as was pointed out in 4 for sufficiently small values of n solutions of 0 1 which are initially strictly positive may vanish at some point x 0 after a finite time to but to the best of our knowledge there is no rigorous proof of this phenomenon in the literature on the other hand it is of particular interest for applications for example if u denotes the thickness of a liquid film it implies that at time to the film breaks at xo

linking anisotropic sharp and diffuse surface motion laws via gradient flows we compare four surface motion laws for sharp surfaces with their diffuse interface counterparts by means of gradient flows on corresponding energy functionals the energy functionals can be defined to give the same dependence on normal direction for the energy of sharp plane surfaces as for their diffuse counterparts the anisotropy of the kinetics can be incorporated into the inner product without affecting the energy functional

quantifying vagueness in decision models abstract when modelling specific decision situations the decision maker often feels overstrained when he is asked for precise numerical quotations concerning the objectives or the constraints whereas qualitative statements are easily given in the recent past the theory of fuzzy sets has proven to be very useful for representing this type of information though it is quite advanced formally the practical determination of its core elements i e membership functions and operators has only been explored to a very limited extent this paper presents results of empirical research which focused on the problem of modelling vagueness i e determining membership functions of fuzzy sets which are considered as quantitative representations of vague concepts such as young man long sticks high profits etc

k3 surfaces of genus 8 and varieties of sums of powers of cubic fourfolds the main result of this paper is that the variety of presentations of a general cubic form f in 6 variables as a sum of 10 cubes is isomorphic to the fano variety of lines of a cubic 4 fold f in general different from f z f a general k3 surface s of genus 8 determines uniquely a pair of cubic 4folds the apolar cubic f s and the dual pfaffian cubic f s or for simplicity f and f as beauville and donagi have shown the fano variety tf of lines on the cubic f is isomorphic to the hilbert scheme hilb2 s of length two subschemes of s the first main result of this paper is that hilb2 s parametrizes the variety vsp f 10 of presentations of the cubic form f with f z f as a sum of 10 cubes which yields an isomorphism between nff and vsp f 10 furthermore we show that vsp f 10 sets up a 6 10 correspondence between f and ff the main result follows by a deformation argument 1 pfaffian and apolar cubic 4 folds associated to k3 surfaces of genus 8 1 1 let v be a 6 dimensional vector space over c fix a basis e e5 for v then ei a ej for 0 i j aijei aej e a2v a skewsymmetric matrix m g aij with aji aj with pliicker coordinates xij the embedding of the grassmannian g g 2 v in p14 p a2v is then precisely the locus of rank 2 skewsymmetric 6 x 6 matrices 0 x01 x02 x03 x04 x05 x0i 0 x12 x13 x14 x15 m x02 x12 0 x23 x24 x25 x03 x13 x23 0 x34 x35 x04 x14 x24 x34 0 x45 x05 x15 x25 x35 x45 0 since the sum of two rank 2 matrices has rank at most 4 and any rank 4 skewsymmetric matrix is the sum of two rank 2 skewsymmetric matrices the secant variety of g is the cubic hypersurface k defined by the 6 x 6 pfaffian m of the matrix m the dual variety of g in p14 p a2v is a cubic hypersurface k k cf 10 k is the secant variety of g g v 2 the grassmannian of rank 2 quotient spaces of v and of course g c g received by the editors july 5 1999 2000 mathematics subject classification primary 14j70 secondary 14m15 14n99 2000 american mathematical society

finite element approximation of the cahn hilliard equation with concentration dependent mobility we consider the cahn hilliard equation with a logarithmic free energy and non degenerate concentration dependent mobility in particular we prove that there exists a unique solution for sufficiently smooth initial data further we prove an error bound for a fully practical piecewise linear finite element approximation in one and two space dimensions finally some numerical experiments are presented

inverse system of a symbolic power ii the waring problem for forms abstract the classical waring problem for forms is to determine the smallest length s of an additive decomposition of a general degree d homogeneous polynomial or form f in r variables as sum of s d th powers of linear forms we show that its solution is implied by a result of j alexander and a hirschowitz concerning the hilbert functions of the ideal of functions vanishing to order two at a generic set of s points in p r 1 using macaulay s inverse systems we show that the alexander hirschowitz result is equivalent to determining the number of linear syzygies of s homogeneous forms in r variables that are d th powers of a given set of general linear forms we also determine the dimension of the family of degree d forms that have additive decompositions of length s we then study several notions of length for forms f having to do with the kind of length s zero dimensional schemes z in p r 1 whose defining ideal i z annihilates the inverse system of f when z is to consist of distinct points we obtain the above length of additive decomposition of f when z is smoothable we obtain the smoothable length of f when z is arbitrary we obtain a scheme length of f all these lengths are at least as large as the dimension of the vector space of all order i partial derivates of f for each i the above mentioned length functions are distinct using results about the existence of nonsmoothable gorenstein point singularities in codimension 4 we show that when r 5 there are forms f of scheme length s which are not in the closure of the family of forms having additive decompositions of length s finally we propose a new set of waring problems for forms using these lengths

unconditionally stable methods for simulating multi component two phase interface models with peng robinson equation of state and various boundary conditions in this paper we consider multi component dynamic two phase interface models which are formulated by the cahn hilliard system with peng robinson equation of state and various boundary conditions these models can be derived from the minimum problems of helmholtz free energy or grand potential in the realistic thermodynamic systems the resulted cahn hilliard systems with various boundary conditions are fully coupled and strongly nonlinear a linear transformation is introduced to decouple the relations between different components and as a result the models are simplified from this we further propose a semi implicit unconditionally stable time discretization scheme which allows us to solve the cahn hilliard system by a decoupled way and thus our method can significantly reduce the computational cost and memory requirements the mixed finite element methods are employed for the spatial discretization and the approximate errors are also analyzed for both space and time numerical examples are tested to demonstrate the efficiency of our proposed methods

on anisotropic order parameter models for multi phase system and their sharp interface limits abstract for a general class of diffuse anisotropic multi phase order parameter or phase field models we use formally matched asymptotic expansions to determine the asymptotic limit when a small parameter related to the thickness of the interface tends to zero in the case of anisotropic allen cahn systems we obtain in the limit that the interface moves by anisotropic mean curvature flow at triple junctions a force balance holds which in the anisotropic case includes shear forces herring torque terms acting normal to the interface we further identify the singular limit of anisotropic cahn hilliard systems

canonical curves with low apolarity abstract let k be an algebraically closed field and let c be a non hyperelliptic smooth projective curve of genus g defined over k since the canonical model of c is arithmetically gorenstein macaulay s theory of inverse systems allows us to associate to c a cubic form f in the divided power k algebra r g 3 in g 2 variables the apolarity ap c of c is the minimal number t of linear form l 1 l t r g 3 needed to write f as the sum of their divided power cubes it is easy to see that ap c g 2 and p de poi and f zucconi classified curves with ap c g 2 when k c in this paper we give a complete characteristic free classification of curves c with apolarity g 1 and g 2

canonical curves and varieties of sums of powers of cubic polynomials in this note we show that the apolar cubic forms associated to codimension 2 linear sections of canonical curves of genus g 11 are special with respect to their presentation as sums of cubes

birational aspects of the geometry of varieties of sums of powers abstract varieties of sums of powers describe the additive decompositions of a homogeneous polynomial into powers of linear forms the study of these varieties dates back to sylvester and hilbert but only few of them for special degrees and number of variables are concretely identified in this paper we aim to understand a general birational behavior of vsp to do this we birationally embed these varieties into grassmannians and prove the rational connectedness of many vsp in arbitrary degrees and number of variables

a second order accurate non linear difference scheme for the n component cahn hilliard system a b s t r a c t we consider a second order conservative nonlinear numerical scheme for the ncomponent cahn hilliard system modeling the phase separation of a n component mixture the scheme is based on a crank nicolson finite difference method and is solved by an efficient and accurate nonlinear multigrid method we numerically demonstrate the second order accuracy of the numerical scheme we observe that our numerical solutions are consistent with the exact solutions of linear stability analysis results we also describe numerical experiments such as the evolution of triple junctions and the spinodal decomposition in a quaternary mixture we investigate the effects of a concentration dependent mobility on phase separation