approximating saddle points as equilibria of differential inclusions abstract motivated by nonlinear programming we study a process involving continuous adjustments of both primal and dual variables nonsmoothness in problem data or simple restrictions on variables make the velocity of the process possibly nonunique the process obtained is shown to converge locally under strict monotonicity conditions examples are given in terms of augmented lagrangians the results are believed to provide a basis for the construction of algorithms

discontinuous galerkin finite element approximation of the cahn hilliard equation with convection the paper is concerned with the construction and convergence analysis of a discontinuous galerkin finite element method for the cahn hilliard equation with convection using discontinuous piecewise polynomials of degree p geq1 and backward euler discretization in time we show that the order parameter c is approximated in the broken rm l infty rm h 1 norm with optimal order cal o h p tau the associated chemical potential w phi c gamma 2 delta c is shown to be approximated with optimal order cal o h p tau in the broken rm l 2 rm h 1 norm here phi c frac 1 4 1 c 2 2 is a quartic free energy function and gamma 0 is an interface parameter numerical results are presented with polynomials of degree p 1 2 3

semi global exponential stability of augmented primal dual gradient dynamics for constrained convex optimization abstract primal dual gradient dynamics that find saddle points of a lagrangian have been widely employed for handling constrained optimization problems building on existing methods we extend the augmented primal dual gradient dynamics aug pdgd to incorporate general convex and nonlinear inequality constraints and we establish its semi global exponential stability when the objective function is strongly convex we also provide an example of a strongly convex quadratic program of which the aug pdgd fails to achieve global exponential stability numerical simulation also suggests that the exponential convergence rate could depend on the initial distance to the kkt point

motion by curvature in generalized cahn allen models the cahn allen model for the motion of phase antiphase boundaries is generalized to account for nonlinearities in the kinetic coefficient relaxation velocity and the coefficient of the gradient free energy the resulting equation is varepsilon 2 u l alpha u varepsilon kappa u raise0 5ex hbox scriptstyle 1 kern 0 1em kern 0 15em lower0 25ex hbox scriptstyle 2 nabla cdot kappa u raise0 5ex hbox scriptstyle 1 kern 0 1em kern 0 15em lower0 25ex hbox scriptstyle 2 nabla u f u wheref is bistable hereu is an order parameter and and are physical quantities associated with the system s free energy and relaxation speed respectively grain boundaries away from triple junctions are modeled by solutions with internal layers when e 1 the classical motion by curvature law for solution layers well known when and are constant is shown by formal asymptotic analysis to be unchanged in form under this generalization the only difference being in the value of the coefficient entering into the relation the analysis is extended to the case when the relaxation time for the process vanishes for a set of values ofu then is infinite for those values

stability and robustness for saddle point dynamics through monotone mappings abstract motivated by recent interest in saddle point dynamics where given a convex in x and concave in y function trajectories follow the steepest descent in x and the steepest ascent in y and where convergence of trajectories to saddle points is desired this note revisits the maximal monotone mapping approach to saddle point dynamics mildly improves one convergence result and proposes new results on the robustness of pointwise asymptotic stability of the set of saddle points the results apply to nonsmooth convex concave functions and constraints and as a special case to projected saddle point dynamics

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

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

solving cone constrained convex programs by differential inclusions a differential inclusion is designed for solving cone constrained convex programs the method is of subgradient projection type it involves projection penalties and lagrangian relaxation nonsmooth data can be accommodated a novelty is that multipliers converge monotonically upwards to equilibrium levels an application to stochastic programming is considered

stable patterns in a viscous diffusion equation we consider a pseudoparabolic regularization of a forward backward nonlinear diffusion equation ut a f u vut motivated by the problem of phase separation in a viscous binary mixture the function f is nonmonotone so there are discontinuous steady state solutions corresponding to arbitrary arrangements of phases we find that any bounded measurable steady state solution u x satisfying f u constant f u x 0 a e is dynamically stable to perturbations in the sense of convergence in measure in particular smooth solutions may achieve discontinuous asymptotic states furthermore stable states need not correspond to absolute minimizers of free energy thus violating gibbs principle of stability for phase mixtures

surface evolution of elastically stressed films under deposition by a diffuse interface model we consider the heteroepitaxial growth of thin films by numerical simulations within a diffuse interface model the model is applicable to describe the self organization of nanostructures the influence of strain surface energies and kinetics on the surface evolution is considered a matched asymptotic analysis shows the formal convergence of an anisotropic viscous cahn hilliard model to a general surface evolution equation the system is solved by adaptive finite elements in three dimensions and in special cases compared with sharp interface models

axisymmetric surface diffusion dynamics and stability of self similar pinchoff the dynamics of surface diffusion describes the motion of a surface with its normal velocity given by the surface laplacian of its mean curvature this flow conserves the volume enclosed inside the surface while minimizing its surface area we review the axisymmetric equilibria the cylinder sphere and the delaunay unduloid the sphere is stable while the cylinder is long wave unstable a subcritical bifurcation from the cylinder produces a continuous family of unduloid solutions we present computations that suggest that the stable manifold of the unduloid forms a separatrix between states that relax to the cylinder in infinite time and those that tend toward finite time pinchoff we examine the structure of the pinchoff showing it has self similar structure using asymptotic numerical and analytical methods in addition to a previously known similarity solution we find a countable set of similarity solutions each with a different asymptotic cone angle we develop a stability theory in similarity variables that selects the original similarity solution as the only linearly stable one and consequently the only observable solution we also consider similarity solutions describing the dynamics after the topological transition

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

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

a numerical study of electro migration voiding by evolving level set functions on a fixed cartesian grid a numerical method for studying migration of voids driven by surface diffusion and electric current in a metal conducting line is developed the mathematical model involves moving boundaries governed by a fourth order nonlinear partial differential equation which contains a nonlocal term corresponding to the electrical field and a nonlinear term corresponding to the curvature numerical challenges include efficient computation of the electrical field with sufficient accuracy to afford fourth order differentiation along the void boundary and to capture singularities arising in topological changes we use the modified immersed interface method with a fixed cartesian grid to solve for the electrical field and the fast local level set method to update the position of moving voids numerical examples are performed to demonstrate the physical mechanisms by which voids interact under electromigration

computation for electromigration in interconnects of microelectronic devices abstract reliability and performance of microelectronic devices depend to a large extent on the resistance of interconnect lines voids and cracks may occur in the interconnects causing a severe increase in the total resistance and even open circuits in this work we analyze void motion and evolution due to surface diffusion effects and applied external voltage the interconnects under consideration are three dimensional sandwich constructs made of a very thin metal film of possibly variable thickness attached to a substrate of nonvanishing conductance a two dimensional level set approach was applied to study the dynamics of the moving assumed one dimensional boundary of a void in the metal film the level set formulation of an electromigration and diffusion model results in a fourth order nonlinear two dimensional time dependent pde this equation was discretized by finite differences on a regular grid in space and a runge kutta integration scheme in time and solved simultaneously with a second order static elliptic pde describing the electric potential distribution throughout the interconnect line the well posed three dimensional problem for the potential was approximated via singular perturbations in the limit of small aspect ratio by a two dimensional elliptic equation with variable coefficients describing the combined local conductivity of metal and substrate which is allowed to vary in time and space the difference scheme for the elliptic pde was solved by a multigrid technique at each time step motion of voids in both weak and strong electric fields was examined and different initial void configurations were considered including circles ellipses polygons with rounded corners a butterfly and long grooves analysis of the void behavior and its influence on the resistance gives the circuit designer a tool for choosing the proper parameters of an interconnect width to length ratio properties of the line material conductivity of the underlayer etc

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

some dynamic properties of volume preserving curvature driven flows this paper is concerned with the following three types of geometric evolution equations the volume preserving mean curvature flow the intermediate surface diffusion flow and the surface diffusion flow important common properties of these flows are the preservation of volume and the decrease of perimeter it is shown in this paper that the intermediate surface diffusion flow can lose convexity hence the volume preserving mean curvature flow is the only flow among the evolution equations under consideration which preserves convexity cf 11 16 14 17 moreover several sufficient conditions are presented which illustrate that each of the above mentioned flows can move smooth initial configurations into singularities in finite time

some mathematical challenges in materials science four challenges to mathematics research posed by the field of materials science are given plus an additional challenge purely from the field of geometric measure theory the problems all concern the effects of surface and grain boundary free energy motion by weighted mean curvature and or surface diffusion and or other kinetics proofs of minimality of soap bubble clusters and their anisotropic analogs shapes of crystals in a gravitational or other field incorporating data from simulations into mathematics and understanding flat chains modulo the figures are selected copies of transparencies presented at the lecture on which this paper is based materials science is both much older and much younger than mathematics ancient humans discovered various elemental metals like gold and copper at some point they discovered that if you mix a little tin into copper you get something quite different from either namely bronze there is a period of human history labeled the bronze age i don t think there is any labeled the arithmetic age on the other hand the term materials science only came into general use within the last 30 years or so as the subjects of metallurgy ceramics and polymers were collected into one discipline the materials research society was founded in 1973 and the international congress of materials research societies in 1991 and along with the old problems of making better structural materials such as steel came new issues unimaginable a century ago some of these include the ability to see and manipulate individual atoms with an atomic force microscope the study of carbon nanotubes the fabrication of electronic chips and the making of thin films for cd s and other storage media what are mathematicians doing in such a field first of all as rodney nichols said at the new york academy of sciences on june 5 2000 much more than a few decades ago there are mathematicians everywhere and that is for the good secondly there has been an effort by the national science foundation to encourage this interdisciplinary research a search of the nsf database under the division of mathematical sciences performed in the year 2000 revealed 85 grants since january 1 1997 with the words materials science in their abstracts there have been several multi million dollar awards co sponsored by darpa for such cooperative efforts targeted at specific problems part of the result is that the collection of mathematicians working on problems from materials science has grown from isolated individuals prior to the mid 1980 s to a small community and now received by the editors november 6 2000 and in revised form february 21 2002 2000 mathematics subject classification primary 74n20 secondary 49q20 49q15 35r99

least uncertainty principle in deformation quantization deformation quantization generally produces families of cohomologically equivalent quantizations of a single physical system we conjecture that the physically meaningful ones i allow enough observable energy distributions i e ones for which no pure quantum state has negative probability and ii reduce the uncertainty in the probability distribution of the resulting quantum states for the simple harmonic oscillator this principle selects the classic groenewold moyal or weyl product on phase space while for the free particle in which there is no real quantization all cohomologically equivalent quantizations are equally good

on finite convergence and constraint identification of subgradient projection methods this paper deals with a continuous time subgradient projection algorithm shown to generate trajectories that accumulate to the solution set under a strong convexity assumption we show that convergence is exponential in norm a sharpness condition yields convergence in finite time and the necessary lapse is estimated invoking a constraint qualification and a non degeneracy assumption we demonstrate that optimally active constraints are identified in finite time