motion of a set by the curvature of its boundary abstract the study of a crystal shrinking or growing in a melt gives rise to equations relating the normal velocity of the motion to the curvature of the crystal boundary often these equations are anisotropic indicating the preferred directions of the crystal structure in the isotropic case this equation is called the curve shortening or the mean curvature flow equation and has been studied by differential geometric tools in general there are no classical solutions to these equations in this paper we develop a weak theory for the generalized mean curvature equation using the newly developed theory of viscosity solutions our approach is closely related to that of osher and sethian chen giga and goto and evans and spruck who view the boundary of the crystal as the level set of a solution to a nonlinear parabolic equation although we use their results in an essential way we give an intrinsic definition our main results are the existence of a solution large time asymptotics of this solution and its connection to the level set solution of osher and sethian chen giga and goto and evans and spruck in general there is no uniqueness even for classical solutions but we prove a uniqueness result under restrictive assumptions we also construct a class of explicit solutions which are dilations of wulff crystals

orthogonality conditions and asymptotic stability in the stefan problem with surface tension we prove nonlinear asymptotic stability of steady spheres in the two phase stefan problem with surface tension our method relies on the introduction of appropriate orthogonality conditions in conjunction with a high order energy method

the hele shaw problem with surface tension in a half plane abstract in this paper we consider the hele shaw problem in a 2 dimensional fluid domain t which is constrained to a half plane the boundary of t consist of two components 0 t which lies on the boundary of the half plane and t which lies inside the half plane on t we impose the classical boundary conditions with surface tension and on 0 t we prescribe the normal derivative of the fluid pressure at the point where 0 t and t meet there is an abrupt change in the boundary condition giving rise to a singularity in the fluid pressure we prove that the problem has a unique solution with smooth free boundary t for some small time interval

the polyharmonic heat flow of closed plane curves abstract in this paper we consider the polyharmonic heat flow of a closed curve in the plane our main result is that closed initial data with initially small normalised oscillation of curvature and isoperimetric defect flows exponentially fast in the c topology to a simple circle our results yield a characterisation of the total amount of time during which the flow is not strictly convex quantifying in a sense the failure of the maximum principle

on an evolution equation in a cell motility model abstract this paper deals with the evolution equation of a curve obtained as the sharp interface limit of a non linear system of two reaction diffusion pdes this system was introduced as a phase field model of crawling motion of eukaryotic cells on a substrate the key issue is the evolution of the cell membrane interface curve which involves shape change and net motion this issue can be addressed both qualitatively and quantitatively by studying the evolution equation of the sharp interface limit for this system however this equation is non linear and non local and existence of solutions presents a significant analytical challenge we establish existence of solutions for a wide class of initial data in the so called subcritical regime existence is proved in a two step procedure first for smooth h 2 initial data we use a regularization technique second we consider non smooth initial data that are more relevant from the application point of view here uniform estimates on the time when solutions exist rely on a maximum principle type argument we also explore the long time behavior of the model using both analytical and numerical tools we prove the nonexistence of traveling wave solutions with nonzero velocity numerical experiments show that presence of non linearity and asymmetry of the initial curve results in a net motion which distinguishes it from classical volume preserving curvature motion this is done by developing an algorithm for efficient numerical resolution of the non local term in the evolution equation

some mathematical aspects of the micro phase separation in diblock copolymers abstract a free energy functional of nonlocal type is considered that was originally introduced to describe the micro phase separation of diblock copolymer a mathematical framework is given to the issues of the scaling law for stationary states and the governing equation of morphology the scaling law is equivalent to finding a nice rescaling in order to have a well defined limiting interfacial problem which is free from the interfacial thickness and the total chain length of copolymer and the associated stationary problem becomes a morphology equation that governs the configuration of final patterns although the steady patterns become finer and finer in our scaling limit due to the mesoscopic nature a possible rigorous approach is presented to stability and morphological selection problems for them

free boundary problems with surface tension conditions abstract in this paper we consider free boundary problems with a surface tension condition at the free boundary the simplest of these problems is the classical hele shaw problem which describes viscous flow between two parallel plates with small spacing we shall consider models which describe tumor growth and are expressed by systems of pdes we shall also consider the hele shaw problem constrained in a half space

an improved algorithm for evaluating trellis phase codes corresp a method is described for evaluating the minimum distance parameters of trellis phase codes including continuous phase frequency shift keying cpfsk partial response fm and more importantly coded continuous phase modulation cpm schemes the algorithm provides dramatically faster execution times and smaller memory requirements than previous algorithms results of sample calculations and timing comparisons are included

a paradifferential approach for well posedness of the muskat problem we study the muskat problem for one fluid or two fluids with or without viscosity jump with or without rigid boundaries and in arbitrary space dimension d of the interface the muskat problem is scaling invariant in the sobolev space h s c mathbb r d where s c 1 frac d 2 employing a paradifferential approach we prove local well posedness for large data in any subcritical sobolev spaces h s mathbb r d s s c moreover the rigid boundaries are only required to be lipschitz and can have arbitrarily large variation the rayleigh taylor stability condition is assumed for the case of two fluids with viscosity jump but is proved to be automatically satisfied for the case of one fluid the starting point of this work is a reformulation solely in terms of the drichlet neumann operator the key elements of proofs are new paralinearization and contraction results for the drichlet neumann operator in rough domains

the hele shaw problem with surface tension in a half plane a model problem abstract consider the hele shaw problem with surface tension in the half plane y 1 0 when at time t 0 the domain t lies partly on the line y 1 0 and partly in y 1 0 in order to establish existence of a solution to this free boundary problem we need to study the linear model problem when the t is a fixed angular domain in this paper we consider this model problem and establish existence of a solution satisfying sharp weighted holder estimates these estimates will be used in subsequent work to solve the full hele shaw problem

ostwald ripening for dilute systems under quasistationary dynamics we consider a dilute mixture in 3d of a finite number of particles initially close to spherical but of varying size evolving according to quasistationary dynamics under the scaling hypotheses that 1 size distance 2 deviation from sphericity size are both initially small we show robustness of the almost spherical shape by associating centers and radii to each particle and then derive rigorously a set of o d e s for the radii which we relate to the lifschitz slyozov wagner theory of coarsening and also establish an estimate for the speed of the centers

convergence of the cahn hilliard equation to the hele shaw model we prove that level surfaces of solutions to the cahn hilliard equation tend to solutions of the hele shaw problem under the assumption that classical solutions of the latter exist the method is based on a new matched asymptotic expansion for solutions a spectral analysis for linearizd operators and an estimate for the difference between the true solutions and certain approximate ones

trellis coding with asymmetric modulations traditionally symmetric signal constellations i e those with uniformly spaced signal points have been used for both uncoded and coded systems although symmetric signal constellations are optimum with no coding the same is not necessarily true for coded systems this paper shows that by designing the signal constellations to be asymmetric one can in many instances obtain a performance gain over the traditional symmetric constellations combined with trellis coding in particular we consider the joint design of n n 1 trellis codes and asymmetric 2 n 1 point signal constellations which has no bandwidth expansion relative to an uncoded 2n point symmetric signal set the asymptotic performance gains due to coding and asymmetry are evaluated in terms of the minimum free euclidean distance d free of the trellis a comparison of the maximum value of this performance measure to the minimum distance d min of the uncoded system is an indication of the maxiamm reduction in required e b n 0 that can be achieved for arbitrarily small system bit error rates bit error probability analysis is carried out for general cases a few examples are given to show the performance gain due to the asymmetry of the signal set it is to be emphasized that the introduction of asymmetry into the signal set does not affect the bandwidth or power requirements of the system hence the abovementioned improvements in performance come at little or no cost asymmetric signal sets in coded systems first appear in the work of divsalar and yuen 1 2 here we expand upon these results by considering various types of asymmetric signal sets combined with the optimum in the sense of maximum d free trellis code having 2 4 8 and 16 states the numerical results obtained will clearly demonstrate the tradeoff between the additional savings in required e b n 0 and the additional complexity more trellis states needed to achieve it

darcy s flow with prescribed contact angle well posedness and lubrication approximation we consider the spreading of a thin two dimensional droplet on a solid substrate we use a model for viscous fluids where the evolution is governed by darcy s law at the contact point where air and liquid meet the solid substrate a constant non zero contact angle partial wetting is assumed we show local and global well posedness of this free boundary problem in the presence of the moving contact point our estimates are uniform in the contact angle assumed by the liquid at the contact point in the so called lubrication approximation long wave limit we show that the solutions converge to the solution of a one dimensional degenerate parabolic fourth order equation which belongs to a family of thin film equations the main technical difficulty is to describe the evolution of the non smooth domain and to identify suitable spaces that capture the transition to the asymptotic model uniformly in the small parameter varepsilon

well posedness and uniform bounds for a nonlocal third order evolution operator on an infinite wedge we investigate regularity and well posedness for a fluid evolution model in the presence of a three phase contact point we consider a fluid evolution governed by darcy s law after linearization we obtain a nonlocal third order operator which contains the dirichlet neumann operator on the wedge with opening angle epsilon 0 we show well posedness and regularity for this linear evolution equation in the limit of vanishing opening angle we show the convergence of solutions to a fourth order degenerate parabolic operator related to the thin film equation in the course of the analysis we introduce and characterize a new type of sum of weighted sobolev spaces which are suitable to capture the singular limit as epsilon to 0 in particular the nature of the problem requires the use of techniques that are adapted to the problem in the singular domain as well as the degenerate limit problem

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

global error estimates for the standard parallel shooting method we present an a posteriori procedure for estimating the global discretization error in the numerical solution of a boundary value problem for a system of m first order ordinary differential equations when the parallel shooting algorithms is used both nonlinear and linear systems are considered and expressions of the corresponding global discretization error are given showing slight formal differences between the two cases as shooting reduces boundary value problems to initial value problems an error estimation for a boundary value problem is obtained assuming that an error estimation procedure for initial value problems is available however the number of initial value problems to be considered in the present error estimation is sharply reduced with respect to the number of initial value problems proposed by the shooting method

approximate solutions of the cahn hilliard equation via corrections to the mullins sekerka motion we develop an alternative method to matched asymptotic expansions for the construction of approximate solutions of the cahn hilliard equation suitable for the study of its sharp interface limit the method is based on the hilbert expansion used in kinetic theory besides its relative simplicity it leads to calculable higher order corrections to the interface motion

using gauss jordan elimination to compute the index generalized nullspaces and drazin inverse abstract we develop and analyze a new algorithm that computes bases for the null spaces of all powers of a given matrix as well as its index the algorithm uses row operations and shuffling steps in which rows of pairs of matrices are interchanged in particular the new algorithm may be viewed as an extension of the classic gauss jordan elimination method for inverting a nonsingular matrix it is also shown that the drazin inverse has a simple representation in terms of the output of the algorithm and the original matrix

well posedness of the muskat problem with h2 initial data abstract we study the dynamics of the interface between two incompressible fluids in a two dimensional porous medium whose flow is modeled by the muskat equations for the two phase muskat problem we establish global well posedness and decay to equilibrium for small h 2 perturbations of the rest state for the one phase muskat problem we prove local well posedness for h 2 initial data of arbitrary size finally we show that solutions to the muskat equations instantaneously become infinitely smooth

the muskat problem with surface tension and a nonregular initial interface abstract in this paper we consider the two dimensional muskat problem with surface tension on a free boundary the initial shape of the unknown interface has a corner point we prove that the problem has a unique solution in the weighted holder classes locally in time and specify the sufficient conditions for the existence of the waiting time phenomenon

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

front migration in the nonlinear cahn hilliard equation the method of matched asymptotic expansions is used to describe solutions of the nonlinear cahn hilliard equation for phase separation in n 1 space dimensions the expansion is formally valid when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature on the dominant slowest timescale the interface velocity is determined by the mean curvature of the interface by a non local relation which is identical to that in a well known quasi static model of solidification which exhibits a shape instability discovered by mullins sekerka j appl phys 34 323 329 1963 on a faster timescale the cahn hilliard equation regularizes a classic two phase stefan problem similarity solutions of the two phase stefan problem should describe boundary layers existence and uniqueness of such similarity solutions which admit metastable states is proved rigorously in an appendix

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

computational studies of coarsening rates for the cahn hilliard equation with phase dependent diffusion mobility we study computationally coarsening rates of the cahn hilliard equation with a smooth double well potential and with phase dependent diffusion mobilities the latter is a feature of many materials systems and makes accurate numerical simulations challenging our numerical simulations confirm earlier theoretical predictions on the coarsening dynamics based on asymptotic analysis we demonstrate that the numerical solutions are consistent with the physical gibbs thomson effect even if the mobility is degenerate in one or both phases for the two sided degenerate mobility we report computational results showing that the coarsening rate is on the order of l c t 1 4 independent of the volume fraction of each phase for the one sided degenerate mobility that is non degenerate in the positive phase but degenerate in the negative phase we illustrate that the coarsening rate depends on the volume fraction of the positive phase for large positive volume fractions the coarsening rate is on the order of l c t 1 3 and for small positive volume fractions the coarsening rate becomes l c t 1 4

on well posedness of the muskat problem with surface tension abstract we consider the muskat problem with surface tension for one fluid or two fluids with or without viscosity jump with infinite depth or lipschitz rigid boundaries and in arbitrary dimension d of the interface the problem is nonlocal quasilinear and to leading order is scaling invariant in the sobolev space h s c r d with s c 1 d 2 we prove local well posedness for large data in all subcritical sobolev spaces h s r d s s c allowing for initial interfaces whose curvatures are unbounded and furthermore when d 1 not locally square integrable to the best of our knowledge this is the first large data well posedness result that covers all subcritical sobolev spaces for the muskat problem with surface tension we reformulate the problem in terms of the dirichlet neumann operator and use a paradifferential approach to reduce the problem to an explicit parabolic equation which is of independent interest

optimized signal constellations for trellis coded modulation on awgn channels previously performance gains over ungerboeck type trellis coded modulation schemes were obtained by optimizing by hand the signal constellation using genetic algorithms and simulated annealing we have found additional cases with performance gains over the ungerboeck type schemes

ostwald ripening in two dimensions the rigorous derivation of the equations from the mullins sekerka dynamics we consider a dilute mixture of a finite number of particles and we are interested in the coarsening of the spatial distribution in two space dimensions under mullins sekerka dynamics under the appropriate scaling hypotheses we associate radii and centers to each particle and derive equations for the whole evolution

the hele shaw problem and area preserving curve shortening motions we prove existence locally in time of a solution of the following hele shaw problem given a simply connected curve contained in a smooth bounded domain find the motion of the curve such that its normal velocity equals the jump of the normal derivatives of a function which is harmonic in the complement of the curve in and whose boundary value on the curve equals its curvature we show that this motion is a curve shortening motion which does not change the area of the region enclosed by the curve in case is the whole plane 2 we also show that if the initial curve is close to an equilibrium curve i e to a circle then there exists a global solution and the global solution tends to some circle exponentially fast as time tends to infinity

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

higher order implicit multistep methods for matrix differential equations abstract this paper proposes implicit multistep matrix methods for the numerical solution of stiff initial value matrix problems the study of matrix difference equations involving the matrix coefficients of the multistep method permits one to obtain convergence results as well as bounds for the global discretization error in terms of the data an illustrative example is included

well posedness and qualitative behaviour of the mullins sekerka problem with ninety degree angle boundary contact we show local well posedness for a mullins sekerka system with ninety degree angle boundary contact we will describe the motion of the moving interface by a height function over a fixed reference surface using the theory of maximal regularity together with a linearization of the equations and a localization argument we will prove well posedness of the full nonlinear problem via the contraction mapping principle here one difficulty lies in choosing the right space for the neumann trace of the height function and showing maximal l p l q regularity for the linear problem in the second part we show that solutions starting close to certain equilibria exist globally in time are stable and converge to an equilibrium solution at an exponential rate

local and global existence results for anisotropic hele shaw flows in this paper we study a moving boundary problem for an anisotropic two phase hele shaw flow using a regularization technique we prove existence of a local solution under suitable conditions on the initial free boundary we obtain a global solution and study its asymptotic behaviour

motion of elastic thin films by anisotropic surface diffusion with curvature regularization in this paper we prove short time existence uniqueness and regularity for a surface diffusion evolution equation with curvature regularization in the context of epitaxially strained two dimensional films this is achieved by using the h 1 gradient flow structure of the evolution law via de giorgi s minimizing movements this seems to be the first short time existence result for a surface diffusion type geometric evolution equation in the presence of elasticity

the transmission problem in domains with a corner point for the laplace operator in weighted holder spaces abstract we prove the existence and uniqueness of a solution to the elliptic transmission problem in nonsmooth domains in the weighted holder space the coercive estimates of the solution are given

lyapunov functions identities and the cauchy problem for the hele shaw equation this article is devoted to the study of the hele shaw equation we introduce an approach inspired by the water wave theory starting from a reduction to the boundary introducing the dirichlet to neumann operator and exploiting various cancellations we exhibit parabolic evolution equations for the horizontal and vertical traces of the velocity on the free surface this allows to quasi linearize the equations in a natural way by combining these exact identities with convexity inequalities we prove the existence of hidden lyapunov functions of different natures we also deduce from these identities and previous works on the water wave problem a simple proof of the well posedness of the cauchy problem the analysis contains two side results of independent interest firstly we give a principle to derive estimates for the modulus of continuity of a pde under general assumptions on the flow secondly we prove and give applications of a convexity inequality for the dirichlet to neumann operator