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

on the poincare friedrichs inequality for piecewise h 1 functions in anisotropic discontinuous galerkin finite element methods the purpose of this paper is to propose a proof for the poincare friedrichs inequality for piecewise h 1 functions on anisotropic meshes by verifying suitable assumptions involved in the newly proposed proof we show that the poincare friedrichs inequality for piecewise h 1 functions holds independently of the aspect ratio which characterizes the shape regular condition in finite element analysis in addition under the maximum angle condition we establish the poincare friedrichs inequality for the crouzeix raviart non conforming linear finite element counterexamples show that the maximum angle condition is only sufficient

analysis and applications of the voronoi implicit interface method we analyze a new mathematical and numerical framework the voronoi implicit interface method viim first introduced in saye and sethian 2011 r i saye j a sethian the voronoi implicit interface method for computing multiphase physics pnas 108 49 2011 19498 19503 for tracking multiple interacting and evolving regions phases whose motion is determined by complex physics fluids mechanics elasticity etc from a mathematical point of view the method provides a theoretical framework for moving interface problems that involve multiple junctions defining the motion as the formal limit of a sequence of related problems discretizing this theoretical framework provides a numerical methodolology which automatically handles multiple junctions triple points and quadruple points in two dimensions as well as triple lines etc in higher dimensions topological changes in the system occur naturally with no surgery required in this paper we present the method in detail and demonstrate several new extensions of the method to different physical phenomena including curvature flow with surface energy densities defined on a per phase basis as well as multiphase fluid flow in which density viscosity and surface tension can be defined on a per phase basis we test this method in a variety of ways we perform rigorous analysis and demonstrate convergence in both two and three dimensions for a variety of evolving interface problems including verification of von neumann mullins law in two dimensions and its analog in three dimensions as well as normal driven flow and curvature flow with and without constraints demonstrating topological change and the effects of different boundary conditions we couple the method to a second order projection method solver for incompressible fluid flow and study the effects of membrane permeability and impermeability large shearing torsional forces and the effects of varying density viscosity and surface tension on a per phase basis finally we demonstrate convergence in both space and time of a topological change in a multiphase foam

high accuracy solutions to energy gradient flows from material science models a computational framework is presented for materials science models that come from energy gradient flows the models of interest lead to the evolution of structure involving two or more phases the framework includes higher order derivative models and vector problems solutions are considered in periodic cells and standard fourier spectral discretization in space is used implicit time stepping is used with adaptivity based on local error estimates the implicit system at every time step is solved iteratively with newton s method the resulting linear systems are solved in inner iterations with the conjugate gradient method using a novel preconditioner that is a constant coefficient version of the system taking values for the coefficients at the pure phase states solutions with high spatial and temporal accuracy are obtained the dependence of the condition number of the preconditioned system on the size of the time step and the order parameter in the model that represents the scaled width of transition layers between phases is investigated numerically and with formal asymptotics in a simple setting the asymptotic results require a conjecture on the rank of a modified square distance matrix results from a fast graphical processing unit implementation for a three dimensional model are shown a comparison to time stepping with operator splitting into convex and concave parts that guarantees energy decrease in the numerical scheme is done

the minimum angle condition for d simplices abstract in this note we present a natural generalization of the minimum angle condition commonly used in the finite element analysis for planar triangulations to the case of simplicial meshes in any space dimension the equivalence of this condition with some other mesh regularity conditions is proved

domain walls in the coupled gross pitaevskii equations a thorough study of domain wall solutions in coupled gross pitaevskii equations on the real line is carried out including existence of these solutions their spectral and nonlinear stability their persistence and stability under a small localized potential the proof of existence is variational and is presented in a general framework we show that the domain wall solutions are energy minimizers within a class of vector valued functions with nontrivial conditions at infinity the admissible energy functionals include those corresponding to coupled gross pitaevskii equations arising in modeling of bose einstein condensates the results on spectral and nonlinear stability follow from properties of the linearized operator about the domain wall the methods apply to many systems of interest and integrability is not germane to our analysis finally sufficient conditions for persistence and stability of domain wall solutions are obtained to show that stable pinning occurs near maxima of the potential thus giving rigorous justification to earlier results in the physics literature

efficient algorithms for diffusion generated motion by mean curvature the problem of simulating the motion of evolving surfaces with junctions according to some curvature dependent speed arises in a number of applications by alternately diffusing and sharpening characteristic functions for each region a variety of motions have been obtained which allow for topological mergings and breakings and produce no overlapping regions or vacuums however the usual finite difference discretization of these methods is often excessively slow when accurate solutions are sought even in two dimensions we propose a new spectral discretization of these diffusion generated methods which obtains greatly improved efficiency over the usual finite difference approach these efficiency gains are obtained in part through the use of a quadrature based refinement technique by integrating fourier modes exactly and by neglecting the contributions of rapidly decaying solution transients indeed numerical studies demonstrate that the new algorithm is often more than 1000 times faster than the usual finite difference discretization our findings are demonstrated on several examples

towards a statistical theory of texture evolution in polycrystals most technologically useful materials possess polycrystalline microstructures composed of a large number of small monocrystalline grains separated by grain boundaries the energetics and connectivity of the grain boundary network play a crucial role in determining the properties of a material across a wide range of scales a central problem in materials science is to develop technologies capable of producing an arrangement of grains a texture that provides for a desired set of material properties one of the most challenging aspects of this problem is to understand the role of topological reconfigurations during coarsening here we propose an upscaling procedure suitable for large complex systems the procedure is based on numerical experimentation combined with stochastic tools and consists of large scale numerical simulations of a system at a microscopic level statistical analysis of the microscopic data and formulation of the model based on stochastic characteristics predicted by the statistical analysis this approach promises to be valuable in establishing the effective model of microstructural evolution in realistic two and three dimensional systems to test ideas we use our upscaling procedure to study the mesoscopic behavior of a reduced one dimensional network of grain boundaries despite the simplicity of its formulation this model exhibits highly nontrivial behavior characterized by growth and disappearance of grain boundaries and develops probability distributions similar to those observed in higher dimensional simulations here we focus on the grain deletion events which are common to all coarsening systems

on the variational approximation of combined second and fourth order geometric evolution equations we present a variational formulation of combined motion by minus the laplacian of curvature and mean curvature flow as well as related flows the proposed scheme covers both the closed curve case and the case of curves that are connected via triple or quadruple junction points or intersect the external boundary on introducing a parametric finite element approximation we prove stability bounds and compare our scheme with existing approaches the presented scheme has very good properties with respect to the equidistribution of mesh points and if applicable area conservation

dissection of the path simplex in r n into n path subsimplices abstract we review properties of acute and non obtuse simplices and of ortho simplices and path simplices dissection of path simplices is considered which leads to a new result generalization of coxeter s trisection of a path tetrahedron into three path subtetrahedra to arbitrary spatial dimension n moreover following earlier results by korotov and k i ek we show that applying this procedure recursively in the proper way leads to a self similar path simplicial refinement towards a chosen vertex of the original path simplex

algorithms for area preserving flows we propose efficient and accurate algorithms for computing certain area preserving geometric motions of curves in the plane such as area preserving motion by curvature these schemes are based on a new class of diffusion generated motion algorithms using signed distance functions in particular they alternate two very simple and fast operations namely convolution with the gaussian kernel and construction of the distance function to generate the desired geometric flow in an unconditionally stable manner we present applications of these area preserving flows to large scale simulations of coarsening

averaging of directional derivatives in vertices of nonobtuse regular triangulations for a shape regular triangulation mathcal t h without obtuse angles of a bounded polygonal domain omega subset re 2 let mathcal l h be the space of continuous functions linear on the triangles from mathcal t h and h the interpolation operator from c overline omega to mathcal l h this paper is devoted to the following classical problem find a second order approximation of the derivative partial u partial z a in a direction z of a function u in c 3 overline omega in a vertex a in the form of a linear combination of the constant directional derivatives partial pi h u partial z on the triangles surrounding a an effective procedure for such an approximation is presented its error is proved to be of the size o h 2 an operator mbox w h mathcal l h longrightarrow mathcal l h times mathcal l h relating a second order approximation wh h u of nabla u to every u in c 3 overline omega is constructed and shown to be a so called recovery operator the accuracy of the presented approximation is compared with the accuracies of the local approximations by other known techniques numerically

longest edge n section algorithms in this paper we survey all known including own recent results properties of the longest edge n section algorithms these algorithms in classical and recently designed conforming form are nowadays used in many applications including finite element simulations computer graphics etc as a reliable tool for controllable mesh generation in addition we present a list of open problems arising in and around this topic

kernels with prescribed surface tension mobility for threshold dynamics schemes abstract we show how to construct a convolution kernel that has a desired anisotropic surface tension and desired anisotropic mobility to be used in threshold dynamics schemes for simulating weighted motion by mean curvature of interfaces including networks of them in both two and three dimensions moreover we discuss necessary and sufficient conditions for the positivity of the kernel which in the case of two phase flow ensures that the resulting scheme respects a comparison principle and implies convergence to the viscosity solution of the level set formulation of the flow in particular we show in a barrier type statement that the kernel cannot possibly be positive unless both the mobility and the surface tension satisfy necessary conditions in three dimensions and give a complete characterization among other results is a threshold dynamics scheme that is guaranteed to dissipate a non local approximation to the interfacial energy in the fully anisotropic multiphase setting using the new kernel construction

stability of basic steady states of networks in bounded domains in this article we study the problem of a network of curves in a planar domain with the normal velocity proportional to each curvature and fixed angle conditions at the points at which the curves intersect we introduce the evolution problem of networks identify two cases of basic steady states in bounded and smooth domains and study their stability in terms of the geometry of the boundary

a double bubble in a ternary system with inhibitory long range interaction the free energy of a ternary system with a self organization property includes an interface energy and a longer ranging inhibitory interaction energy in a planar domain if the two energies are properly balanced and two of the three constituents make up an equal but small fraction the free energy admits a local minimizer that is shaped like a perturbed double bubble most difficulties in the proof of this result are related to the triple junction phenomenon that the three constituents of the ternary system meet at a point two techniques are developed to deal with the triple junction first one defines restricted classes of perturbed double bubbles each perturbed double bubble in a restricted class is obtained from a standard double bubble by a special perturbation the two triple junction points of the standard double bubble can only move along the line connecting them in opposite directions and by the same distance the second technique is the use of the so called internal variables these variables derive from the more geometric quantities that describe perturbed double bubbles in restricted classes the advantage of the internal variables is that they are only subject to linear constraints and perturbed double bubbles in a restricted class represented by internal variables are elements of a hilbert space a local minimizer of the free energy in each restricted class is found as a fixed point of a nonlinear equation by a contraction mapping argument the second variation at the fixed point within its restricted class is positive this perturbed double bubble satisfies three of the four equations for critical points of the free energy the unsolved equation is the 120 degree angle condition at triple junction points then perform another minimization among the local minimizers from all restricted classes a minimum of minimizers emerges and solves all the equations for critical points

the bernstein von mises theorem and nonregular models we study the asymptotic behaviour of the posterior distribution in a broad class of statistical models where the true solution occurs on the boundary of the parameter space we show that in this case the bayesian inference is consistent and that the posterior distribution has not only gaussian components as in the case of regular models the bernstein von mises theorem but also has gamma distribution components that depend on the behaviour of the prior distribution on the boundary and have a faster rate of convergence we also show a remarkable property of bayesian inference that for some models there appears to be no bound on efficiency of estimating the unknown parameter if it is on the boundary of the parameter space we illustrate the results on a problem from emission tomography

inference and testing on the boundary in extended constant conditional correlation garch models we consider inference and testing in extended constant conditional correlation garch models in the case where the true parameter vector is a boundary point of the parameter space this is of particular importance when testing for volatility spillovers in the model the large sample properties of the qmle are derived together with the limiting distributions of the related lr wald and lm statistics due to the boundary problem these large sample properties become nonstandard the size and power properties of the tests are investigated in a simulation study as an empirical illustration we test for no volatility spillovers between foreign exchange rates

generalized transition waves and their properties in this paper we generalize the usual notions of waves fronts and propagation speeds in a very general setting these new notions which cover all usual situations involve uniform limits with respect to the geodesic distance to a family of hypersurfaces that are parametrized by time we prove the existence of new such waves for some time dependent reaction diffusion equations as well as general intrinsic properties some monotonicity properties and some uniqueness results for almost planar fronts the classification results which are obtained under some appropriate assumptions show the robustness of our general definitions 2012 wiley periodicals inc

a front dynamics approach to curvature dependent flow a front dynamics approach is developed to study the evolution of planar curves whose normal speed depends on curvature the formulation is similar to whitham s shock dynamics theory for the propagation of shock wave in gases but assumes a different propagation rule equations that describe the motion of the front are obtained and these are evolution equations for the normal direction and local arc length of the front the solution of these equations leads to the front positions using an appropriate integration along rays a similarity solution of the equations is found for the evolution of an initial corner free boundary problems for the motion of a junction connecting front segments are discussed a numerical method is presented to calculate the evolution of any number of front segments the segments can be closed or open connected to wall boundaries or not or connected to other segments at 3 segment junctions several sample problems are considered to illustrate the method an extension of the metho

slow motion in higher order systems and g convergence in one space dimension the slow motion in higher order systems and convergence in one space dimension was studied some of the theorems and computations that support and proves the stability of the equations are presented

a variational approach to multiple layers of the bistable equation in long tubes multiple layer solutions of the balanced bistable equation in infinite tubes are constructed via a variational method i start with a characterization of palais smale sequences which easily gives some global minima in the desired function classes as single layers assuming these minima are isolated as critical points i paste them together to serve as an approximate multiple layer solution if there were no exact solutions near the approximate one the negative gradient flow of the energy functional would significantly lower the energy on the other hand if the minima are kept far from each other the energy of a function near the approximate solution is not much less than that of the approximate solution this contradiction proves the existence of a solution

grain boundary motion arising from the gradient flow of the aviles giga functional abstract this paper considers the singular limit of the equation t 2 1 2 1 grain boundaries limiting discontinuities in form networks that coarsen over time a matched asymptotic analysis is used to derive a free boundary problem consisting of curve motion coupled along hyperbolic characteristics and junction conditions an intermediate boundary layer near extrema junctions is discovered along with the relevant nonlocal junction conditions the limiting dynamics can be viewed in the context of a gradient flow of the sharp interface energy on an attracting manifold dynamic scaling of the long time coarsening process can be explained by dimensional analysis of the reduced problem

metastable bubble solutions for the allen cahn equation with mass conservation in a multidimensional domain the slow motion behavior of internal layer solutions with spherical interfaces referred to as bubble solutions is analyzed for the nonlocal allen cahn equation with mass conservation this problem represents the simplest model for the phase separation of a binary mixture in the presence of a mass constraint the bubble is shown to drift exponentially slowly across the domain without change of shape toward the closest point on the boundary of the domain an explicit ordinary differential equation for the motion of the center of the bubble is derived by extending to a multidimensional setting the asymptotic projection method developed previously by the author to treat metastable problems in one spatial dimension an asymptotic formula for the time of collapse of the bubble against the boundary of the domain is derived in terms of the principal radii of curvature of the boundary at the initial contact point an analogy between slow bubble motion and the classical exit prob

diffusion generated motion using signed distance functions we describe a new class of algorithms for generating a variety of geometric interfacial motions by alternating two steps construction of the signed distance function i e redistancing to the interface and convolution with a suitable kernel these algorithms can be seen as variants of merriman bence and osher s threshold dynamics 25 the new algorithms proposed here preserve the computational efficiency of the original threshold dynamics algorithm however unlike threshold dynamics the new algorithms also allow attaining high accuracy on uniform grids without adaptive refinement

higher energy solutions in the theory of phase transitions a variational approach abstract we establish the existence of a higher energy solution to the vector ginzburg landau equation with a triple well potential on a bounded and smooth domain on the plane this solution is obtained by a linking argument in implementing this variational approach we make several considerations on the dynamics of the negative gradient flow in particular we use the conley index to contruct a suitable one dimensional invariant set this solution has morse index two in the nondegenerate case we discuss its structure in connection with the so called triple junction configurations

numerical simulation and linear well posedness analysis for a class of three phase boundary motion problems we investigate the linear well posedness for a class of three phase boundary motion problems and perform some numerical simulations in a typical model three phase boundaries evolve under certain evolution laws with specified normal velocities the boundaries meet at a triple junction with appropriate conditions applied a system of partial differential equations and algebraic equations pdae is proposed to describe the problems with reasonable assumptions all problems are shown to be well posed if all three boundaries evolve under the same evolution law for problems involving two or more evolution laws we show the well posedness case by case for some examples numerical simulations are performed for some examples

on the asymptotics of constrained m estimation limit theorems for an m estimate constrained to lie in a closed subset of r d are given under two different sets of regularity conditions a consistent sequence of global optimizers converges under chernoff regularity of the parameter set a n consistent sequence of local optimizers converges under clarke regularity of the parameter set in either case the asymptotic distribution is a projection of a normal random vector on the tangent cone of the parameter set at the true parameter value limit theorems for the optimal value are also obtained agreeing with chernoff s result in the case of maximum likelihood with global optimizers

hamiltonian identities for elliptic partial differential equations abstract new identities for elliptic partial differential equations are obtained several applications are discussed in particular young s law for the contact angles in triple junction formation is proven rigorously structure of level curves of saddle solutions to allen cahn equation are also carefully analyzed

a variational method for multiphase volume preserving interface motions we develop a numerical method for realizing mean curvature motion of interfaces separating multiple phases whose volumes are preserved throughout time the foundation of the method is a thresholding algorithm of the bence merriman osher type the original algorithm is reformulated in a vector setting which allows for a natural inclusion of constraints even in the multiphase case moreover a new method for overcoming the inaccuracy of thresholding methods on non adaptive grids is designed since this inaccuracy becomes especially prominent in volume preserving motions formal analysis of the method and numerical tests are presented

on the heteroclinic connection problem for multi well gradient systems abstract we revisit the existence problem of heteroclinic connections in r n associated with hamiltonian systems involving potentials w r n r having several global minima under very mild assumptions on w we present a simple variational approach to first find geodesics minimizing length of curves joining any two of the potential wells where length is computed with respect to a degenerate metric having conformal factor w then we show that when such a minimizing geodesic avoids passing through other wells of the potential at intermediate times it gives rise to a heteroclinic connection between the two wells this work improves upon the approach of 22 and represents a more geometric alternative to the approaches of e g 5 10 14 17 for finding such connections

diffusion generated motion for grain growth in two and three dimensions an efficient algorithm for accurately simulating curvature flow for large networks of curves in two dimensions and surfaces in three dimensions on uniform grids is proposed this motion arises in the technologically important problem of simulating grain boundary motion in polycrystalline materials in this formulation grain boundaries are zero level sets of signed distance functions curvature motion is achieved by first diffusing locally maintained signed distance functions followed by a reinitialization step a technique is devised to allow a single signed distance function to represent a large subset of spatially separated grains hundreds of thousands of grains can be simulated using a small number of signed distance functions in this work 32 in two dimensions and 64 in three dimensions are more than sufficient using modest computational hardware

a multiphase field concept numerical simulations of moving phase boundaries and multiple junctions we present numerical simulations which support the formal asymptotic analysis relating a multiorder parameter allen cahn system to a multiphase interface problem with curvature dependent evolution of the interfaces and angle conditions at triple junctions within the gradient energy of the allen cahn system the normal to an interface between phases i and j is modeled by the irreducible representations u i nabla u j u j nabla u i u i nabla u j u j nabla u i where ui and uj are the ith and jth components of the vectorial order parameter bu in rz n in the vectorial case the dependence of the limiting surface tensions and mobilities on the bulk potentials of the allen cahn system is not given explicitly but in terms of all the n components of the planar stationary wave solutions one of the issues of this paper is to find bulk potentials which allow a rather easy access to the resulting surface tensions and mobilities we compare numerical computations for planar and circular phase bound

a parametric finite element method for fourth order geometric evolution equations we present a finite element approximation of motion by minus the laplacian of curvature and related flows the proposed scheme covers both the closed curve case and the case of curves that are connected via triple junctions on introducing a parametric finite element approximation we prove stability bounds and compare our scheme with existing approaches it turns out that the new scheme has very good properties with respect to area conservation and the equidistribution of mesh points we state also an extension of our scheme to willmore flow of curves and discuss possible further generalizations

a novel element differential method for solid mechanical problems using isoparametric triangular and tetrahedral elements abstract a novel strong form numerical method element differential method edm is developed to solve geometrically complex mechanics problems based on triangular or tetrahedral meshes the discretization of the structure under investigation has been based on lagrange isoparametric quadrilateral or hexahedral elements while applying edm in this paper a new family of isoparametric triangular and tetrahedral elements with a central node is proposed for edm a set of shape functions with analytical expressions for their first and second order partial derivatives is constructed for these triangular and tetrahedral elements respectively moreover a new element collocation scheme is proposed to establish a system of equations directly from the governing differential equations for internal nodes and traction equilibrium equations for nodes on edges of an element in this collocation scheme no variational principles or virtual energy principles are required to set up the solution scheme while no integration is needed when forming the coefficients of the system of equations numerical examples including standard patch tests and more practical problems are given to demonstrate the correctness of the constructed elements and the efficiency of the proposed element collocation method

traveling wave solutions of a gradient system solutions with a prescribed winding number ii on considere un systeme d equations de la forme u t u xx f u u r 2 on definit un nombre d enroulement pour les solutions on demontre que pour chaque entier positif k il existe au moins deux solutions onde de propagation chacune ayant un nombre d enroulement k ou k 1

on variational characterization of four end solutions of the allen cahn equation in the plane abstract in this paper a novel variational method is developed to construct four end solutions in r 2 for the allen cahn equation four end solutions have been constructed by del pino kowalczyk pacard and wei when the angle of the ends is close to 2 or 0 using the lyapunov schmid reduction method and later by kowalczyk liu and pacard using a continuation method for general 0 2 by a special mountain pass argument in a restricted space namely a set of monotone paths of monotone functions a family of solutions in bounded domain is constructed with very good control of their nodal sets which then leads to a four end solution after sending the domains to r 2 in this way not only a family of four end solutions is constructed for any angle 0 2 the morse index of such a four end solution is also shown to be one

acute type refinements of tetrahedral partitions of polyhedral domains we present a new technique to perform refinements on acute type tetrahedral partitions of a polyhedral domain provided that the center of the circumscribed sphere around each tetrahedron belongs to the tetrahedron the resulting family of partitions is of acute type thus all the tetrahedra satisfy the maximum angle condition both these properties are highly desirable in finite element analysis

qualitative properties of discrete nonlinear parabolic operators this paper is devoted to the qualitative properties of discretized parabolic operators such as nonnegativity and nonpositivity preservation maximum minimum principles and maximum norm contractivity in the linear case earlier papers of the authors farago and horvath in siam sci comput 28 2313 2336 2006 ima j numer anal 29 606 631 2009 have established the connections between the above qualitative properties and have given sufficient conditions for their validity the present paper extends the above results to nonlinear discretized parabolic operators also motivated by the authors recent paper farago and horvath in j math anal appl 448 473 497 2017 which has given related results on the continuous pde level a systematic study is presented ranging from general discrete mesh operators to proper finite element applications

fast reaction slow diffusion and curve shortening the reaction diffusion problem u 1 varepsilon delta u varepsilon 1 v n u quad u x 0 varepsilon g x quad partial n u 0 text on partial omega for a vector u x t varepsilon is considered in a domain omega in r m an asymptotic solution is constructed for varepsilon small it shows that at each x u tends quickly to a minimum of v u when v has several minima when u tends to a piecewise constant function boundary layer expansions are constructed around the resulting surfaces of discontinuity or fronts each front is found to move along its normal with a constant velocity determined by the discontinuity v in v across it when v 0 the front s normal velocity is varepsilon kappa where kappa is its mean curvature the motion of fronts in this manner is studied for arcs in the plane which are normal to partial omega at their endpoints and for fronts that are closed curves it is shown a front can shrink to a point in

asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions abstract large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described specifically the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics are derived these results generalize the work of moran 1971 chant 1974 and chernoff 1954 some of chant s results are shown to be incorrect the approach used in deriving these results follows from comments made by moran and chant the problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate gaussian distribution from a sample of size 1 in this representation the gaussian random variable corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the normalized maximum likelihood estimator thus the limiting distribution of the maximum likelihood estimator is the same as the distribution of the projection of the gaussian random v

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

on the maximum angle condition for linear tetrahedral elements synge s maximum angle condition for triangular elements is generalized to tetrahedral elements for the generalized condition it is proved that tetrahedra may degenerate in a certain way and the error of the standard linear interpolation remains o h in the w p 1 omega norm for sufficiently smooth functions and p in 1 infty

multiphase thermomechanics with interfacial structure 2 evolution of an isothermal interface paper 1 1988 g 1 of this series began an investigation whose goal is a thermomechanics of two phase continua based on gibbs s notion of a sharp phase interface endowed with thermomechanical structure in that paper a balance law balance of capillary forces was introduced and then applied in conjunction with suitable statements of the first two laws of thermodynamics the chief results are thermodynamic restrictions on constitutive equations exact and approximate free boundary conditions at the interface and a hierarchy of free boundary problems the simplest versions of these problems the mullins sekerka problems are essentially the classical stefan problem with the free boundary condition u 0 for the temperature replaced by the condition u h k where k is the curvature of the free boundary and h 0 is a material constant this dependence on curvature renders the problem difficult and apart from numerical studies involving linearization stability there are almost no supporting theoretical results

minimizers and gradient flows for singularly perturbed bi stable potentials with a dirichlet condition minimizers and gradient flows are studied for the functional w u 2 u 2 d x r n 0 where u satisfies a dirichlet condition u h on here w is taken to be a double well potential with minimum value zero attained at u a and u b questions of existence and structure of minimizers for small are resolved through the identification of a limiting variational problem the so called limit a formal asymptotic solution is then constructed for the gradient flow t u 2 u 1 w9 u u x 0 g x u x t h on valid when is small using multiple timescales we show that fronts rapidly develop and then propagate with normal velocity k where k is mean curvature at the intersection of a front with the dirichlet condition is shown to imply a contact angle condition for the front this asymptotically correct evolution process represents gradient flow for the limit

phase transitions and generalized motion by mean curvature we study the limiting behavior of solutions to appropriately rescaled versions of the allen cahn equation a simplified model for dynamic phase transitions we rigorously establish the existence in the limit of a phase antiphase interface evolving according to mean curvature motion this assertion is valid for all positive time the motion interpreted in the generalized sense of evans spruck and chen giga goto after the onset of geometric singularities

on global and local mesh refinements by a generalized conforming bisection algorithm we examine a generalized conforming bisection gcb algorithm which allows both global and local nested refinements of the triangulations without generating hanging nodes it is based on the notion of a mesh density function which prescribes where and how much to refine the mesh some regularity properties of generated sequences of refined triangulations are proved several numerical tests demonstrate the efficiency of the proposed bisection algorithm it is also shown how to modify the gcb algorithm in order to generate anisotropic meshes with high aspect ratios

error estimates of triangular finite elements under a weak angle condition in this note by analyzing the interpolation operator of girault and raviart given in v girault p a raviart finite element methods for navier stokes equations theory and algorithms in springer series in computational mathematics springer verlag berlin 1986 over triangular meshes we prove optimal interpolation error estimates for lagrange triangular finite elements of arbitrary order under the maximal angle condition in a unified and simple way the key estimate is only an application of the bramble hilbert lemma

heteroclinic connections for multiple well potentials the anisotropic case we investigate the existence of solutions to systems of n differential equations representing connections between minima of potentials with several equal depths in r using variational techniques and in particular a method introduced in af we first prove such existence for n 2 and two minima dealing next with symmetric potentials corresponding to free bulk energies in crystals we establish existence for n 2 in various cases of more than two minima finally we obtain a sufficient condition establishing existence of connections to not necessarily symmetric potentials for arbitrary n and three minima

on the asymptotics of constrained local m estimators we discuss in this paper asymptotics of locally optimal solutions of maximum likelihood and more generally m estimation procedures in cases where the true value of the parameter vector lies on the boundary of the parameter set s we give a counterexample showing that regularity of s in the sense of clarke is not sufficient for asymptotic equivalence of n consistent locally optimal m estimators we argue further that stronger properties such as so called near convexity or prox regularity of s are required in order to ensure that any two n consistent locally optimal m estimators have the same asymptotics

on the equivalence of regularity criteria for triangular and tetrahedral finite element partitions in this note we examine several regularity criteria for families of simplicial finite element partitions in r d email protected 2 3 these are usually required in numerical analysis and computer implementations we prove the equivalence of four different definitions of regularity often proposed in the literature the first one uses the volume of simplices the others involve the inscribed and circumscribed ball conditions and the minimal angle condition

the existence problem for steiner networks in strictly convex domains we consider the existence problem for steiner networks trivalent graphs with 2 3 angles at each junction in strictly convex domains with neumann boundary conditions for each of the three possible combinatorial possibilities sufficient conditions on the domain are derived for existence in addition in each case explicit examples of nonexistence are given

sensitivity analysis of parameterized variational inequalities in this paper we discuss local uniqueness continuity and differentiability properties of solutions of parameterized variational inequalities generalized equations to this end we use two types of techniques one approach consists in formulating variational inequalities in a form of optimization problem based on regularized gap functions and applying a general theory of perturbation analysis of parameterized optimization problems another approach is based on a theory of contingent outer graphical derivatives and some results about differentiability properties of metric projections

an algebraic discrete maximum principle in hilbert space with applications to nonlinear cooperative elliptic systems discrete maximum principles are derived for finite element discretizations of nonlinear elliptic systyems with cooperative and weakly diagonally dominant coupling the results are achieved via an algebraic discrete maximum principle organized in a hilbert space framework and in the case of simplicial elements are obtained under weakened acute type conditions for the finite element fe meshes

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

multiphase solutions to the vector allen cahn equation crystalline and other complex symmetric structures we present a systematic study of entire symmetric solutions u mathbb r n rightarrow mathbb r m of the vector allen cahn equation delta u w u u 0 quad text for all quad x in mathbb r n where w mathbb r m rightarrow mathbb r is smooth symmetric nonnegative with a finite number of zeros and where w u partial w partial u 1 dots partial w partial u m top we introduce a general notion of equivariance with respect to a homomorphism f g rightarrow gamma g gamma reflection groups and prove two abstract results concerning the cases of g finite and g discrete for the existence of equivariant solutions our approach is variational and based on a mapping property of the parabolic vector allen cahn equation and on a pointwise estimate for vector minimizers

entire solutions to equivariant elliptic systems with variational structure we consider the system u wu u 0 where u mathbb r n to mathbb r n for a class of potentials w mathbb r n to mathbb r that possess several global minima and are invariant under a general finite reflection group g we establish existence of nontrivial g equivariant entire solutions connecting the global minima of w along certain directions at infinity

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

existence and differentiability of metric projections in hilbert spaces this paper considers metric projections onto a closed subset s of a hilbert space if the set s is convex then it is well known that the corresponding metric projections always exist unique and directionally differentiable at boundary points of s these properties of metric projections are considered for possibly nonconvex sets s in particular existence and directional differentiability of metric projections for certain classes of sets are established and will be referred to as nearly convex sets

the maximum angle condition is not necessary for convergence of the finite element method we show that the famous maximum angle condition in the finite element analysis is not necessary to achieve the optimal convergence rate when simplicial finite elements are used to solve elliptic problems this condition is only sufficient in fact finite element approximations may converge even though some dihedral angles of simplicial elements tend to

a numerical method for tracking curve networks moving with curvature motion abstract a finite difference method is proposed to track curves whose normal velocity is given by their curvature and which meet at different types of junctions the prototypical example is that of phase interfaces that meet at prescribed angles although eutectic junctions and interactions through nonlocal effects are also considered the method is based on a direct discretization of the underlying parabolic problem and boundary conditions a linear stability analysis is presented for our scheme as well as computational studies that confirm the second order convergence to smooth solutions after a singularity in the curve network where the solution is no longer smooth we demonstrate almost second order convergence a numerical study of singularity types is done for the case of networks that meet at prescribed angles at triple junctions finally different discretizations and methods for implicit time stepping are presented and compared

threshold dynamics for networks with arbitrary surface tensions we present and study a new algorithm for simulating the n phase mean curvature motion for an arbitrary set of isotropic surface tensions the departure point is the threshold dynamics algorithm of merriman bence and osher for the two phase case a new energetic interpretation of this algorithm allows us to extend it in a natural way to the case of n phases for arbitrary surface tensions and arbitrary isotropic mobilities for a large class of surface tensions the algorithm is shown to be consistent in the sense that at every time step it decreases an energy functional that is an approximation in the sense of gamma convergence of the interfacial energy a broad range of numerical tests shows good convergence properties an important application is the motion of grain boundaries in polycrystalline materials it is also established that the above mentioned large class of surface tensions contains the read shockley surface tensions both in the two dimensional and three dimensional settings 2015 wiley periodicals inc

travelling wave solutions to a gradient system soit un systeme d equations de reaction diffusion ou la non linearite derive d un potentiel avec certaines restrictions on utilise l indice de conley et la matrice de connexion pour montrer qu il y a une solution onde de propagation reliant les maxima du potentiel

on three phase boundary motion and the singular limit of a vector valued ginzburg landau equation 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

local differentiability of distance functions recently clarke stern and wolenski characterized in a hilbert space the closed subsets c for which the distance function dc is continuously differentiable everywhere on an open tube of uniform thickness around c here a corresponding local theory is developed for the property of dc being continuously differentiable outside of c on some neighborhood of a point x c this is shown to be equivalent to the prox regularity of c at x which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation additional characterizations are provided in terms of dc being locally of class c 1 or such that dc 2 is convex around x for some 0 prox regularity of c at x corresponds further to the normal cone mapping nc having a hypomonotone truncation around x and leads to a formula for pc by way of nc the local theory also yields new insights on the global level of the clarke stern wolenski results and on a property of sets introduced by shapiro as well as on the concept of sets with positive reach considered by federer in the finite dimensional setting

a diffusion generated approach to multiphase motion in this article we present a diffusion generated approach for evolving multiple junctions this work generalizes an earlier method by merriman bence and osher which alternately diffuses and sharpens characteristic functions for each phase region to produce pure mean curvature flow specifically our new method produces a normal velocity which depends on a positive multiple of the curvature of the interface plus the difference in bulk energy densities for prescribed junction angles this simple method naturally treats topological mergings and breakings produces no overlapping regions or vacuums and can be made very fast numerical studies are provided which show that our method agrees with front tracking and a recent variational approach for a variety of examples asymptotic expansions are also carried out near junctions to justify our algorithms

global and local refinement techniques yielding nonobtuse tetrahedral partitions preservation of basic qualitative properties for example the maximum principle ofthe solution of partial differential equations by its finite element approximations is an important goal in mathematical modelling and simulation nonobtuse tetrahedral partitions and linear finite elements guarantee the validity of the discrete analogues of the maximum principle for a wide class of parabolic and elliptic problems in order to get more accurate approximation we often need to refine the used partitions globally or locally in this paper we first propose two variants of global refinement techniques which produce nonobtuse face to face tetrahedral partitions second we present a new local refinement technique which generates nonobtuse face to face tetrahedral partitions in a neighbourhood of a given vertex