analysis of a variable time step discretization for the penrose fife phase relaxation problem in this paper we study a fully implicit time discrete scheme with variable time steps for a system of field equations of penrose fife type governing the dynamics of phase transitions with a nonconserved order parameter in particular fourier law for the heat flux and zero interfacial energy are assumed uniform positivity existence and uniqueness of the discrete solution is proved several a priori estimates are estabilished then passage to the limit is reached via compactness and monotonicity arguments optimal order estimates for the discretization error are shown

front motion for phase transitions in systems with memory abstract we consider the following partial integro differential equation allen cahn equation with memory 2 t 0 t a t t 2 f h t d t where is a small parameter h a constant f the negative derivative of a double well potential and the kernel a is a piecewise continuous differentiable at the origin scalar valued function on 0 the prototype kernels are exponentially decreasing functions of time and they reduce the integro differential equation to a hyperbolic one the damped klein gordon equation by means of a formal asymptotic analysis we show that to the leading order and under suitable assumptions on the kernels the integro differential equation behaves like a hyperbolic partial differential equation obtained by considering prototype kernels the evolution of fronts is governed by the extended damped born infeld equation we also apply our method to a system of partial integro differential equations which generalize the classical phase field equations with a non conserved order parameter and describe the process of phase transitions where memory effects are present u t 2 t 0 t a 1 t t u t d t 2 t 0 t a 2 t t 2 f u t d t where is a small parameter in this case the functions u and represent the temperature field and order parameter respectively the kernels a1 and a2 are assumed to be similar to a for the phase field equations with memory we obtain the same result as for the generalized klein gordon equation or allen cahn equation with memory

asymptotic behaviour of a phase field model with three coupled equations without uniqueness abstract we prove the existence of weak solutions for a phase field model with three coupled equations with unknown uniqueness and state several dynamical systems depending on the regularity of the initial data then the existence of families of global attractors level set depending for the corresponding multi valued semiflows is established applying an energy method finally using the regularizing effect of the problem we prove that these attractors are in fact the same

υ preinvexity and υ invexity in mathematical programming in the paper we introduce and characterize a new class of nonconvex functions in mathematical programming named c preinvex functions which are called c invex functions in the case of differentiability the class of c reinvex functions is wider than that of preinvex functions and thus in the case of differentiability than the class of invex functions some optimality results are obtained under c preinvexity assumption for a constrained optimization problem with not necessarily differentiable functions further a number of sufficiency conditions and wolfe duality theorems are established under c invexity assumption an alternative approach with a modified c invexity notion to optimality conditions and duality results is also considered

the determination of the poles of the mapping function and their use in numerical conformal mapping abstract let be the function which maps conformally a simply connected domain onto the unit disc this paper is concerned with the problem of determining the dominant poles of in compl and of using this information in order to obtain accurate numerical approximations to by means of the bergman kernel method

p r invex sets and functions abstract notions of invexity of a function and of a set are generalized the notion of an invex function with respect to can be further extended with the aid of p invex sets slight generalization of the notion of p invex sets with respect to leads to a new class of functions a family of real functions called in general p r pre invex functions with respect to without differentiability or p r invex functions with respect to in the differentiable case is introduced some geometric properties of these classes of functions are derived sufficient optimality conditions are obtained for a nonlinear programming problem involving p r invex functions with respect to

thin interface asymptotics for an energy entropy approach to phase field models with unequal conductivities karma and rappel phys rev e 57 1998 4342 recently developed a new sharp interface asymptotic analysis of the phase field equations that is especially appropriate for modeling dendritic growth at low undercoolings their approach relieves a stringent restriction on the interface thickness that applies in the conventional asymptotic analysis and has the added advantage that interfacial kinetic effects can also be eliminated however their analysis focussed on the case of equal thermal conductivities in the solid and liquid phases when applied to a standard phase field model with unequal conductivities anomalous terms arise in the limiting forms of the boundary conditions for the interfacial temperature that are not present in conventional sharp interface solidification models as discussed further by almgren siam j appl math 59 1999 2086 in this paper we apply their asymptotic methodology to a generalized phase field model which is derived using a thermodynamically consistent approach that is based on independent entropy and internal energy gradient functionals that include double wells in both the entropy and internal energy densities the additional degrees of freedom associated with the generalized phase field equations can be used to eliminate the anomalous terms that arise for unequal conductivities

well posedness for solid liquid phase transitions with a fourth order nonlinearity abstract a phase field system which describes the evolution of both the absolute temperature and the phase variable f during first order transitions in thermal insulators is considered a thermodynamic approach is developed by regarding the order parameter as a phase field and its evolution equation as a balance law by virtue of the special form of the internal energy a third order nonlinearity g 2 f appears in the energy balance in place of the customarily constant latent heat as a consequence the bounds 0 f 1 hold whenever is positive valued in addition a nonlinear fourier law with conductivity proportional to temperature is assumed well posedness for the resulting initial and boundary value problem are then established in a suitable setting

error estimates for a finite element discretization of a phase field model for mixtures we derive error estimates for finite element discretizations of phase field models that describe phase transitions in nonisothermal mixtures special attention is paid to the applicability of the result for a large class of models with nonlinear constitutive relations and to an approach that avoids an exponential dependence of the constants in the error estimate on the approximation parameter that models the thickness of the diffuse phase transition region the main assumptions on the model are a convexity condition for a function that can be interpreted as the negative local part of the entropy of the system a suitable regularity of the exact solutions and a spectrum estimate for the operator of the allen cahn equation the spectrum estimate is crucial to avoid the exponential dependence of error constants on the approximation parameters in the model this is done by a technique introduced in x feng and a prohl math comp 73 2004 pp 541 567 for phase transitions of pure materials with linear constitutive relations

retrieving topological information for phase field models the phase field approach has become a popular tool in modeling interface motion microstructure evolution and more recently the shape transformation of vesicle membranes under elastic bending energy while it is advantageous to employ phase field models in numerical simu lations to automatically handle topological changes to the microstructures or the configurations of vesicle membranes detecting topological events may also become important for many applications such as those in the simulation of blood cells motivated by such considerations a new quantity is formulated to retrieve some topological information based on the phase field formulation and to capture the occurrence of topological events it can also be used as a control method to avoid unphys ical changes of topology due to the numerical methods should it become necessary for particular practical applications through numerical experiments we demonstrate the effectiveness and the robustness of the new quantity in detecting the topology of fluid bubbles and vesicle membranes

well posedness results and asymptotic behaviour for a phase transition model taking into account microscopic accelerations we obtain an existence and uniqueness result to fremond s phase transition model which take into account microscopic movements and accelerations moreover the irreversible evolution of the phase variable is considered next we perform an asymptotic analysis on the solution to the above problem as the power of the microscopic acceleration forces goes to zero

on a coupled bulk surface allen cahn system with an affine linear transmission condition and its approximation by a robin boundary condition abstract we study a coupled bulk surface allen cahn system with an affine linear transmission condition that is the trace values of the bulk variable and the values of the surface variable are connected via an affine relation and this serves to generalize the usual dynamic boundary conditions we tackle the problem of well posedness via a penalization method using robin boundary conditions in particular for the relaxation problem the strong well posedness and long time behaviour of solutions can be shown for more general and possibly nonlinear relations new difficulties arise since the surface variable is no longer the trace of the bulk variable and uniform estimates in the relaxation parameter are scarce nevertheless weak convergence to the original problem with affine linear relations can be shown using the approach of colli and fukao 2015 we show strong existence to the original problem with affine linear relations and derive an error estimate between solutions to the relaxed and original problems

continuity of the temperature in the two phase stefan problem the classical two phase stefan problem a model for ice water melting gives rise to a singular nonlinear partial differential equation which admits a unique weak solution here we prove that this solution and therefore the temperature in the stefan problem are continuous

on a navier stokes allen cahn model with inertial effects abstract a mathematical model describing the flow of two phase fluids in a bounded container is considered under the assumption that the phase transition is influenced by inertial effects the model couples a variant of the navier stokes system for the velocity u with an allen cahn type equation for the order parameter relaxed in time in order to introduce inertia at the microscopic level the resulting model is characterized by second order material derivatives which constitute the main difficulty in the mathematical analysis in order to obtain a tractable problem a viscous relaxation term is then included in the phase equation the mathematical results consist in existence of weak solutions in 3d and under additional assumptions existence and uniqueness of strong solutions in 2d a partial characterization of the long time behavior of solutions is also given and in particular some issues related to dissipation of energy are discussed

a fast algorithm for linear complex chebyshev approximations we propose a new algorithm for finding best minimax polynomial approximations in the complex plane the algorithm is the first satisfactory generalization of the well known remez algorithm for real approximations among all available algorithms ours is the only quadratically convergent one numerical examples are presented to illustrate rapid convergence

convergence of the solution of the stochastic 3d globally modified cahn hilliard navier stokes equations abstract we study in this article the stochastic 3d globally modified cahn hilliard navier stokes model in a 3d dimensional bounded domain we prove the existence and uniqueness of strong solutions furthermore we discuss the relation of the stochastic 3d globally modified cahn hilliard navier stokes equations to the stochastic 3d cahn hilliard navier stokes equations by proving a convergence theorem that as the parameter n tends to infinity a subsequence of solutions of the stochastic 3d globally modified cahn hilliard navier stokes equations converges to a weak martingale solution of the stochastic 3d cahn hilliard navier stokes equations

lyapunov function and spectrum comparison for a reaction diffusion system with mass conservation abstract we are dealing with a two component system of reaction diffusion equations with mass conservation in a bounded domain with the neumann boundary conditions we prove the global boundedness of the solution in l norm for t 0 under a condition and then the existence of a lyapunov function moreover by studying the linearized eigenvalue problem of a nonconstant equilibrium solution we provide a comparison theorem for the spectrum between the linearized operators of the system and an appropriate nonlocal scalar equation as an application of the comparison result we obtain that any stable equilibrium solution must be monotone if the space dimension is one it is also shown that a modified system with a new parameter which covers the present model possesses a lyapunov function

thermodynamically consistent models of phase field type for the kinetics of phase transitions abstract a general framework is given for the phenomenological kinetics of phase transitions in which not only the order parameter but also the temperature may vary in time and space instead of a ginzburg landau free energy functional as used in formulating the cahn hilliard equation we use the analogous entropy functional model entropy functionals and the kinetic equations resulting from them are constructed for various cases phase transitions with and without a critical point and in the former case with or without a latent heat the class considered is general enough to include the entropy functionals for the ising model in mean field approximation the van der waals fluid and a simplified version of the density functional theory of freezing a case without critical point for which the energy is conserved but the order parameter is not provides a thermodynamically consistent derivation of the phase field equations studied by caginalp fix and others and also leads in a natural way to the lyapunov functional given by langer for these equations but the treatment also suggests that a modified version of the phase field equations might provide a more realistic model of freezing

weak solution of a stochastic 3d cahn hilliard navier stokes model driven by jump noise abstract we investigate a stochastic 2d and 3d cahn hilliard navier stokes system with a multiplicative noise of levy type the model consists of the navier stokes equations for the velocity coupled with a cahn hilliard system for the order phase parameter we prove that the system the existence of weak martingale solution for both 2d and 3d cases the proof of the existence is based on a classical galerkin approximation as well as some compactness methods in the 2d case we prove the pathwise uniqueness of the weak solution

existence of global solutions to the caginalp phase field system with dynamic boundary conditions and singular potentials our aim in this article is to prove the global in time existence of solutions to a caginalp phase field system with dynamic boundary conditions and a singular potential the main difficulty is to prove that the solutions are strictly separated from the singular values of the potential this is achieved by studying an auxiliary elliptic problem

the phase field method in the sharp interface limit the phase field pf method for solidification phenomena is an open formulation based on a free energy functional two common choices for the pf potential here referred to briefly as the caginalp and kobayashi models are compared with respect to their numeric results within the classical sharp interface limit both qualitative and quantitative behavior are addressed and an assessment of the computational effort required to approximate a sharp interface problem is made it is shown that the specific form of the free energy potential does have a strong influence on the convergence of the pf results to their sharp interface limit compliance of the pf solutions with the linear kinetic model for the interface temperature is also investigated a simple one dimensional solidification problem in the presence of kinetic undercooling is solved by the pf model and also by a deforming grid method our results support the view that if care is exercised in formulating the phase temperature coupling there is a high degree of confidence in using the pf method for the numerical modeling of general solidification phenomena

the quasistationary phase field equations with neumann boundary conditions abstract we prove that the quasistationary phase field equations t u u f 2e 1 e w u where w t t 2 1 2 is a double well potential admit a solution when the space dimension n 3 and that the solutions converge for e 0 to solutions of the stefan problem with gibbs thomson law

an eulerian description of fluids containing visco elastic particles equations governing the flow of fluid containing visco hyperelastic particles are developed in an eulerian framework the novel feature introduced here is to write an evolution equation for the strain it is envisioned that this will simplify numerical codes which typically compute the strain on lagrangian meshes moving through eulerian meshes existence results for the flow of linear visco hyperelastic particles in a newtonian fluid are established using a galerkin scheme

phase field theory of edges in an anisotropic crystal in the presence of sufficiently strong surface energy anisotropy the equilibrium shape of an isothermal crystal may include corners or edges models of edges have to date involved the regularization of the corresponding free boundary problem resulting in equilibrium shapes with smoothed out edges in this paper we take a new approach and consider how a phase field model which provides a diffuse description of an interface can be extended to the consideration of edges by an appropriate regularization of the underlying mathematical model using the method of matched asymptotic expansions we develop an approximate solution which corresponds to a smoothed out edge from which we are able to determine the associated edge energy

some convergence results for a class of nonlinear phase field evolution equations two heat diffusion problems in the framework of the parabolic phase field model are presented the first problem is related to a single isotropic fluid and the other describes the heat transmission between two different substances in contact some known existence and uniqueness results are briefly recalled then an asymptotic analysis of both situations is carried out as the kinetic equation collapses to a temperature phase relation of stefan type in the first case in the whole material and in the second in only one of the substances in both cases a convergence result for the solutions is proved the second problem shows some more mathematical difficulties that are due to the presence of nontrivial terms on the common boundary in order to control the latter some tools are used from the convergence theory for convex functionals

an analysis of a phase field model of a free boundary a mathematical analysis of a new approach to solidification problems is presented a free boundary arising from a phase transition is assumed to have finite thickness the physics leads to a system of nonlinear parabolic differential equations existence and regularity of solutions are proved invariant regions of the solution space lead to physical interpretations of the interface a rigorous asymptotic analysis leads to the gibbs thompson condition which relates the temperature at the interface to the surface tension and curvature

two numerical methods for the conformal mapping of simply connected domains abstract two numerical methods are considered for the conformai mapping of a bounded simply connected domain onto the unit disc the two methods are respectively the bergman kernel method which has been described in 17 and the so called ritz method in this paper we indicate the close theoretical relationship of the two methods compare their computational efficiencies and present a number of practical applications of the approximate conformal maps

shape instabilities and pattern formation in solidification a new method for numerical solution of the moving boundary problem abstract when a solidification front is advancing into a region of supercooled liquid its shape is subject to instabilities which can lead to complex modes of growth the evaluation of the growth pattern is determined by the interaction of the driving force of the instability due to heat diffusion with the restabilizing force due to surface tension and its full development is a highly nonlinear process a new method for solving the two phase heat diffusion problem under these conditions is described the method is based on a single domain treatment weak formulation of the moving boundary problem with the temperature field near the interface approximated by a piecewise linear function in the spirit of the finite element technique calculated results for a model problem are in excellent agreement with the predictions of an analytical approximation

reconstruction of two time independent coefficients in an inverse problem for a phase field system abstract in this paper we present stability results concerning the inverse problem of determining two time independent coefficients for a phase field system in a bounded domain r n for the dimension n 3 with a single observation on a subdomain and the sobolev norm of certain partial derivatives of the solutions at a fixed positive time 0 t over the whole spatial domain the proof of these results relies on an appropriate carleman estimate for the phase field system

the gibbs thompson relation within the gradient theory of phase transitions this paper discusses the asymptotic behavior as 0 of the chemical potentials associated with solutions of variational problems within the van der waals cahn hilliard theory of phase transitions in a fluid with free energy per unit volume given by 2 2 w where is the density the main result is that is asymptotically equal to e d o with e the interfacial energy per unit surface area of the interface between phases the constant sum of principal curvatures of the interface and d the density jump across the interface this result is in agreement with a formula conjectured by m gurtin and corresponds to the gibbs thompson relation for surface tension proved by g caginalp within the context of the phase field model of free boundaries arising from phase transitions

universal attractor for some singular phase transition systems a singular parabolic system describing the thermal diffusion in a substance possibly subject to a phase transition is introduced the physical process is described by the variables absolute temperature and order parameter the latter may have or not conserved total mass with respect to time in both cases after recalling and sometimes improving some known well posedness results the long time behavior of the system is studied it is shown that the process is dissipative and the compact universal attractor is constructed it turns out to attract the trajectories of the system in a rather strong metric which is strictly linked to the constraints imposed to both variables the techniques used in the proofs seem likely to be applied to other types of evolution systems containing maximal monotone nonlinearities

quasilinear systems with memory kernels we consider a quasilinear integrodifferential system in non normal form such a system is a generalization of a phase field model with memory and includes as a particular case the system describing the combustion of a material with memory in this paper we study both the direct and the inverse problems our fundamental tools are the theory of analytic semigroups optimal regularity results and fixed point arguments

exponential attractors for a conserved phase field system with memory we consider a conserved phase field system of caginalp type characterized by the assumption that both the internal energy and the heat flux depend on the past history of the temperature and its gradient respectively the model consists of a parabolic integrodifferential equation coupled with a fourth order evolution equation for the phase field this system supplemented with suitable boundary conditions can be interpreted as a dissipative dynamical system in a proper phase space in a previous joint work the last two authors have proved the existence of a global attractor of finite fractal dimension here we show the existence of an exponential attractor by means of the techniques we developed for nonconserved phase field systems with memory

the treatment of corner and pole type singularities in numerical conformal mapping techniques abstract this paper is a report of recent developments concerning the nature and the treatment of singularities that affect certain numerical conformal mapping techniques the paper also includes some new results on the nature of singularities that the mapping function may have in the complement of the closure of the domain under consideration

direct and inverse problems for a phase field model with memory we consider a semilinear integrodifferential system in non normal form such a system is a generalization of the one that arises in the phase field theory with memory we prove an abstract existence and uniqueness theorem and a continuous dependence result for the direct problem reformulating the direct problem in a suitable way we prove that the identification problem admits a unique solution

necessary and sufficient conditions in constrained optimization additional conditions are attached to the kuhn tucker conditions giving a set of conditions which are both necessary and sufficient for optimality in constrained optimization under appropriate constraint qualifications necessary and sufficient conditions are also given for optimality of the dual problem duality and converse duality are treated accordingly

solving complex approximation problems by semiinfinite finite optimization techniques a study on convergence we transform a complex approximation problem into an equivalent semiinfinite optimization problem whose constraints are described in terms of a quantity 0 2 i we study the effect of disturbing the problem by replacingi by a compact subsetm i which includes as special case the discrete case wherem consists only of finitely many points we introduce a measure for the deviation ofm fromi and show that in any complex approximation problem the minimal distance of the disturbed problem converges quadratically with 0 to the minimal distance of the undisturbed problem which is a generalization of a result by streit and nuttall we also show that in a linear finite dimensional approximation problem the convergence of the coefficients of the disturbed problem is in general at most linear there are some graphical representations of best complex approximations computed with the described method

the numerical computation of heteroclinic connections in systems of gradient partial differential equations the numerical computation of heteroclinic connections in partial differential equations pdes with a gradient structure such as those arising in the modeling of phase transitions is considered initially a scalar reaction diffusion equation is studied structural assumptions are made on the problem to ensure the existence of absorbing sets and consequently a global attractor as a result of the gradient structure it is known that if all equilibria are hyperbolic the global attractor comprises the set of equilibria and heteroclinic orbits connecting equilibria to one another thus it is natural to consider direct approximation of the set of equilibria and the connecting orbits results are proved about the fourier spanning basis for branches of equilibria and also for certain heteroclinic connections these results exploit the oddness of the nonlinearity the reaction diffusion equation is then approximated by a galerkin spectral discretization to produce a system of ordinary differential equations

steady states solutions of allen cahn equation by computer algebra in this paper we consider the analytical solution and numerical approximation of the celebrated steady states allen cahn equation we present a computer code to solve and plot the solutions using maxima software

optimality and duality for proper and isolated efficiencies in multiobjective optimization we use some advanced tools of variational analysis and generalized differentiation such as the nonsmooth version of fermat s rule the limiting mordukhovich subdifferential of maximum functions and the sum rules for the frechet subdifferential and for the limiting one to establish necessary conditions for local properly efficient solutions and local isolated minimizers of a multiobjective optimization problem involving inequality and equality constraints sufficient conditions for the existence of such solutions are also provided under assumptions of local convex affine functions or l invex infine functions defined in terms of the limiting subdifferential of locally lipschitz functions in addition we propose a type of wolfe dual problems and examine weak strong duality relations under l invexity infineness hypotheses

abstract approach to evolution equations of phase field type and applications a general system of abstract nonlinear parabolic equations deriving from phase field models of heat transfer in anisotropic fluids with phase transitions is studied through a backward finite differences approximation scheme and existence and uniqueness theorems are proved then applications are presented to two physical situations first a standard heat diffusion problem is briefly discussed mainly as a justification of the choices of the abstract setting second a more interesting transmission problem between two different adjoining fluids is studied under very general compatibility hypotheses and the abstract results are adapted to provide an existence and uniqueness theorem also in this case moreover through convex analysis techniques it is performed a mathematical study of the subdifferential type relations linking the phase fields and the temperatures to the internal energies of the fluids email schimper dragon ian pv cnr it keywords phase field models subdifferential operators abstract evolution equations

global existence and asymptotic behavior for the radially symmetric case of a two phase stefan problem with interfacial energy abstract this paper is concerned with global existence in time and asymptotic behavior for the radially symmetric case of a stefan problem with surface tension effects on the interface according to the static gibbs thomson law these problems arise in phase change theory

local minimisers and singular perturbations we construct local minimisers to certain variational problems the method is quite general and relies on the theory of convergence the approach is demonstrated through the model problem it is shown that in certain nonconvex domains n and for e small there exist nonconstant local minimisers u e satisfying u e 1 except in a thin transition layer the location of the layer is determined through the requirement that in the limit u e u 0 the hypersurface separating the states u 0 1 and u 0 1 locally minimises surface area generalisations are discussed with for example vector valued u and anisotropic perturbations replacing u 2

on sufficiency of the kuhn tucker conditions if all the functions fo x f x f x are convex on c then these conditions are also sufficient various classes of functions have been defined for the purpose of weakening this limitation of convexity in mathematical programming mangasarian 11 has speculated that pseudo convexity of x and quasiconvexity off x are the weakest conditions that can be imposed so that the above conditions are sufficient for optimality it will be shown that there are other wide classes of functions for which the conditions are sufficient 545

on a nonlocal phase field system abstract we study the existence uniqueness and continuous dependence on initial data of the solution to a nonlocal phase field system on a bounded domain the system is a gradient flow for a free energy functional with nonlocal interaction also we study the asymptotic behavior of the solution and show the existence of an absorbing set in some metric space

dynamics of layered interfaces arising from phase boundaries the dynamics of a material in two phases is studied in the context of phase field models based on a landau ginzburg free energy functional they consist of a system of two nonlinear diffusion equat

front tracking finite element method for dendritic solidification 765 abstract a two dimensional finite element method capable of tracking sharp interfaces is developed the method is based on a fixed mesh of bilinear isoparametric elements and is extremely simple and easy to use the interfaces are tracked with a set of marker points that define their position at all times several different approaches to finding the interface position and velocity are discussed and their effect on the convergence rate is examined it is shown through numerical examples that the approximation to the interface position converges with second order accuracy in two dimensional problems as opposed to previously developed finite difference algorithms that are only first order accurate the approximation to the interface velocity is shown to be of first order a number of examples are examined including several that provide a careful comparison with previously published results

phase field method to optimize dielectric devices for electromagnetic wave propagation we discuss a phase field method for shape optimization in the context of electromagnetic wave propagation the proposed method has the same functional capabilities as the level set method for shape optimization the first advantage of the method is the simplicity of computation since extra operations such as re initialization of functions are not required the second is compatibility with the topology optimization method due to the similar domain representation and the sensitivity analysis structural shapes are represented by the phase field function defined in the design domain and this function is optimized by solving a time dependent reaction diffusion equation the artificial double well potential function used in the equation is derived from sensitivity analysis we study four types of 2d or 2 5d axisymmetric optimization problems two are the classical problems of photonic crystal design based on the bloch theory and photonic crystal wave guide design and two are the recent topics of designing dielectric left handed metamaterials and dielectric ring resonators

on the relationship between supplemental balances in two theories for pure interface motion a matched asymptotic analysis is used to exhibit the connection between supplemental balance equations arising in sharp interface and phase field theories for transformations between two rigid phases distinguished only by their constant free energy densities the analysis exposes the relationship between the forces and balances arising in the two theories

uniform approximation as a numerical tool for constructing conformal maps abstract let f 0 be the conformal mapping of a given simply connected region r onto a disk d it is known that f 0 is the unique minimizer of f f where f varies in a set h 1 of holomorphic functions defined on r and is the uniform chebyshev norm in this paper approximations to f 0 are computed numerically with the aid of approximation and optimization techniques the essential feature is the incorporation of the geometry of r into the numerical part of this problem several numerical examples are given with figures in addition it has been shown in another paper by one of the authors that by changing slightly one can also produce conformal mappings of r onto prescribed regions other than d an example of this kind involving a triangle is also given

phase transitions with semi diffuse interfaces in this paper we examine new phase field models with semi diffuse interfaces these models have the property that the 1 1 planar phase transitions take place over a finite interval the models also support multiple interface solutions with interfaces centered at arbitrary points l sub 1 l sub 2 l sub n these solutions correspond to local minima of an entropy functional rather than saddle points and are dynamically stable the classical models have no such exact solutions but they do support solutions with n equally spaced transition points where the order parameter transitions between valves p sub min n and p sub max n satisfying 1 p sub min n 0 p sub max n 1 these solutions of the classical model are saddle points of the entropy functional associated with those models and are not dynamically stable

hypersurfaces with mean curvature given by an ambient sobolev function we consider n hypersurfaces j with interior ej whose mean curvature are given by the trace of an ambient sobolev function uj w 1 p r h j uj ej on j 0 1 where ej denotes the inner normal of j we investigate 0 1 when j weakly as varifolds and prove that is an integral n varifold with bounded first variation which still satisfies 0 1 for uj u ej e p has to satisfy

phase field models for hypercooled solidification abstract properties of the solidification front in a hypercooled liquid so called because the temperature of the resulting solid is below the melting temperature are derived using a phase field diffuse interface model certain known properties for hypercooled fronts in specific materials are reflected within our theories such as the presence of thin thermal layers and the trend towards smoother fronts with less pronounced dendrites when the undercooling is increased within the hypercooled regime both an asymptotic analysis to derive the relevant free boundary problems and a rigorous determination of the inner profile of the diffusive interface are given of particular interest is the incorporation of anisotropy and general microscale interactions leading to higher order differential operators these features necessitate a much richer mathematical analysis than previous theories anisotropic free boundary problems are derived from our models the simplest of which involves determining the evolution of a set a solid particle whose boundary moves with velocity depending on its normal vector considerable attention is given to the identification of surface tension to comparison with previous theories and to questions of stability

a microscopic derivation of macroscopic sharp interface problems involving phase transitions macroscopic free boundary problems involving phase transitions e g the classical stefan problem or its modifications are derived in a unified way from a hamiltonian based on a general set of microscopic interactions a hamiltonian of the form x x j x x x x leads to differential equations as a result of fourier transforms expanding the fourier transform ofj in powers ofq the wave number one can truncate the series at anarbitrary orderm and thereby obtainmth order differential equations an asymptotic analysis of these equations in various scalings of the physical parameters then implies limits which are the standard macroscopic models for the dynamics of phase boundaries

hyperbolic phase field dynamics with memory abstract we consider a non conserved phase field system which consists of two nonlinearly coupled hyperbolic integrodifferential equations this model is derived from two basic assumptions the heat flux law is of gurtin pipkin type and the response of the order parameter to the variation of the free energy functional is delayed these hypotheses might be a reasonable attempt to describe for instance the melt of he 4 crystals a suitable initial and boundary value problem is then associated with the system and its well posedness is analyzed in detail

multiobjective programming under d invexity abstract in this paper a generalization of convexity is considered in the case of nonlinear multiobjective programming problem where the functions involved are nondifferentiable by considering the concept of pareto optimal solution and substituting d invexity for convexity the fritz john type and karush kuhn tucker type necessary optimality conditions and duality in the sense of mond weir and wolfe for nondifferentiable multiobjective programming are given

on invex sets and preinvex functions abstract in this paper we consider the class of the invex function introduced by hanson we show that under certain condition an invex function defined on an invex set a is preinvex on a similarly a quasiinvex function defined on an invex set a is pre quasiinvex

computation of sharp phase boundaries by spreading the planar and spherically symmetric cases abstract the sharp interface that arises from any of the major transition problems classical or modified stefan etc can be smoothed using the phase field approach as a numerical tool the basic idea is that the thickness of the interface can be regarded as a mathematical free parameter which can be stretched beyond its physical value for computational convenience the computations in one dimensional space and n dimensions with radial symmetry indicate that this is an efficient method for dealing with stiff equations and results in a very accurate interface determination without explicit tracking the question of optimizing the interfacial thickness with respect to grid size is also considered empirically the technique also provides a numerical verification of the concept of an unstable critical radius of solidification

optimality and duality in nondifferentiable multiobjective optimization involvingd type i and related functions abstract in this paper new classes of functions namely d type i d quasitype i and d pseudo type i are defined for a multiobjective nondifferentiable programming problem kuhn tucker type necessary and sufficient optimality conditions are obtained for a feasible point to be a weak minimum for this problem two duals are formulated and various duality results are given by using the above defined classes of functions considering the concept of a weak minimum

efficient gluing of numerical continuation and a multiple solution method for elliptic pdes numerical continuation calculations for ordinary differential equations odes are by now an established tool for bifurcation analysis in dynamical systems theory as well as across almost all natural and engineering sciences although several excellent standard software packages are available for odes there are for good reasons no standard numerical continuation toolboxes available for partial differential equations pdes which cover a broad range of different classes of pdes automatically a natural ach to this problem is to look for efficient gluing computation approaches with independent components developed by researchers in numerical analysis dynamical systems scientific computing and mathematical modeling in this paper we shall study several elliptic pdes lane emden fowler lane emden fowler with microscopic force caginalp via the numerical continuation software pde2path and develop a gluing component to determine a set of starting solutions for the continuation by exploiting the variational structures of the pdes in particular we solve the initialization problem of numerical continuation for pdes via a minimax algorithm to find multiple unstable solution furthermore for the caginalp system we illustrate the efficient gluing link of pde2path to the underlying mesh generation and the fem matlab pdetoolbox even though the approach works efficiently due to the high level programming language and without developing any new algorithms we still obtain interesting bifurcation diagrams and directly applicable conclusions about the three elliptic pdes we study in particular with respect to symmetry breaking in particular we show for a modified lane emden fowler equation with an asymmetric microscopic force how a fully connected bifurcation diagram splits up into c shaped isolas on which localized pattern deformation appears towards two different regimes we conclude with a section on future software development issues that would be helpful to be addressed to simplify interfaces to allow for more efficient time saving gluing computation for dynamical systems analysis of pdes in the near future

pre invex functions in multiple objective optimization and called such functions convexlike if s is a convex set and if f is a convex function then clearly f is convexlike elster and nehse obtained a saddlepoint optimality condition for convexlike mathematical programs hayashi and komiya 9 also considered convexlike functions and developed a gordan type theorem of the alternative involving convexlike functions and in addition considered lagrangian duality for convexlike programs hanson 7 considered differentiable functions f s r for which there exists an n dimensional vector function q q u such that for all x u e s

on the configurational force balance in thermomechanics the aim of this paper is a new formulation of the configurational force balance in the general framework of thermomechanics to this end invariance requirements of the configurational working are used following an approach proposed by gurtin to define the notion of working in a more general framework the ideas of green and naghdi concerning the basic postulates of thermomechanics are invoked and used consistently the main results are that the configurational forces balance in the general setting of thermomechanics without the invocation of constitutive relations and the pseudo momentum equation for classical and dissipationless thermoelasticity finally we give an application to fracture theory we show that in classical thermoelasticity the derived pseudo momentum balance provides a crack driving force consistent with the well known expression of the energy release rate

d invexity and optimality conditions abstract in this paper a generalization of convexity called d invexity is introduced substituting d invex for convex we get some optimality conditions for nondifferentiable multiobjective programming the application is demonstrated by an example

optimality conditions and duality for nondifferentiable multiobjective programming problems involving d r type i functions in this paper new classes of nondifferentiable functions constituting multiobjective programming problems are introduced namely the classes of d r type i objective and constraint functions and moreover the various classes of generalized d r type i objective and constraint functions are defined for directionally differentiable multiobjective programming problems sufficient optimality conditions and various mond weir duality results are proved for nondifferentiable multiobjective programming problems involving functions of such type finally it is showed that the introduced d r type i notion with r 0 is not a sufficient condition for wolfe weak duality to hold these results are illustrated in the paper by suitable examples