convolution generated motion and generalized huygens principles for interface motion a physical interface can often be modeled as a surface that moves with a velocity determined by the local geometry accordingly there is great interest in algorithms that generate such geometric interface motion in this paper we unify and generalize two simple algorithms for constant and mean curvature based interface motion the classical huygens principle and diffusion generated motion we show that the resulting generalization can be viewed both geometrically as a type of huygens principle and algebraically as a convolution generated motion using the geometric algebraic duality from the unification we construct specific convolution generated motion algorithms for a common class of anisotropic curvature dependent motionlaws we validate these algorithms with numerical experiments and show that they can be implemented accurately and efficiently with adaptive resolution and fast fourier transform techniques

an efficient threshold dynamics method for wetting on rough surfaces the threshold dynamics method developed by merriman bence and osher mbo is an efficient method for simulating the motion by mean curvature flow when the interface is away from the solid boundary direct generalization of mbo type methods to the wetting problem with interfaces intersecting the solid boundary is not easy because solving the heat equation in a general domain with a wetting boundary condition is not as efficient as it is with the original mbo method the dynamics of the contact point also follows a different law compared with the dynamics of the interface away from the boundary in this paper we develop an efficient volume preserving threshold dynamics method for simulating wetting on rough surfaces this method is based on minimization of the weighted surface area functional over an extended domain that includes the solid phase the method is simple stable with o n log n complexity per time step and is not sensitive to the inhomogeneity or roughness of the solid boundary

sub riemannian heat kernels and mean curvature flow of graphs abstract we introduce a sub riemannian analogue of the bence merriman osher algorithm merriman et al 1992 42 and show that it leads to weak solutions of the horizontal mean curvature flow of graphs over sub riemannian carnot groups the proof follows the nonlinear semi group theory approach originally introduced by l c evans 1993 27 in the euclidean setting and is based on new results on the relation between sub riemannian heat flows of characteristic functions of subgraphs and the horizontal mean curvature of the corresponding graphs

remarks on the global sobolev inequalities in the minkowski space rn 1 on demontre des inegalites de sobolev globales sur l espace de minkowski r n 1 le theoreme principal est valable sur des tranches a temps fixe

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

a numerical method for computing minimal surfaces in arbitrary dimension in this paper we propose a numerical method for computing minimal surfaces with fixed boundaries the level set method is used to evolve a codimension 1 surface with fixed codimension 2 boundary in r n under mean curvature flow for n 3 the problem has been approached in d l chopp 1993 and l t cheng d l chopp computing minimal surfaces via level set curvature flow j comput phys 106 1 1993 77 91 and l t cheng the level set method applied to geometrically based motion materials science and image processing ucla cam report 00 20 using the level set method but with a more complicated boundary conditions the method we present can be generalized straightforward to arbitrary dimension and the framework in which it is presented is dimension independent examples are shown for n 2 3 4

threshold dynamics for the piecewise constant mumford shah functional we propose an efficient algorithm for minimizing the piecewise constant mumford shah functional of image segmentation it is based on the threshold dynamics of merriman bence and osher for evolving an interface by its mean curvature we show that a very fast minimization can be achieved by alternating the solution of a linear parabolic partial differential equation and simple thresholding

existence and uniqueness for anisotropic and crystalline mean curvature flows an existence and uniqueness result up to fattening for crystalline mean curvature flows with forcing and arbitrary convex mobilities is proven this is achieved by introducing a new notion of solution to the corresponding level set formulation such a solution satisfies the comparison principle and a stability property with respect to the approximation by suitably regularized problems the results are valid in any dimension and for arbitrary possibly unbounded initial closed sets the approach accounts for the possible presence of a time dependent bounded forcing term with spatial lipschitz continuity as a by product of the analysis the problem of the convergence of the almgren taylor wang minimizing movements scheme to a unique up to fattening flat flow in the case of general possibly crystalline anisotropies is settled

short time uniqueness results for solutions of nonlocal and non monotone geometric equations we describe a method to show short time uniqueness results for viscosity solutions of general nonlocal and non monotone second order geometric equations arising in front propagation problems our method is based on some lower gradient bounds for the solution these estimates are crucial to obtain regularity properties of the front which allow to deal with nonlocal terms in the equations applications to short time uniqueness results for the initial value problems for dislocation type equations asymptotic equations of a fitzhugh nagumo type system and equations depending on the lebesgue measure of the fronts are presented

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

nonlocal second order geometric equations arising in tomographic reconstruction abstract in this paper we study new nonlocal geometric equations which are related to tomographic reconstruction when using the level set approach we treat two additional difficulties which make the work original on one hand the level lines do not evolve along normal directions and the nonlocal term is not of convolution type on the other hand the speed is not necessarily bounded compared to the nonlocal term we prove an existence and uniqueness result for our equation

uniform bound of sobolev norms of solutions to 3d nonlinear wave equations with null condition abstract this article concerns the time growth of sobolev norms of classical solutions to the 3d quasi linear wave equations with the null condition given initial data in h s h s 1 with compact supports the global well posedness theory has been established independently by klainerman 13 and christodoulou 3 respectively for a relatively large integer s however the highest order sobolev energy namely the h s energy of solutions may have a logarithmic growth in time in this paper we show that the h s energy of solutions is also uniformly bounded for s 5 the proof employs the generalized energy method of klainerman enhanced by weighted l 2 estimates and the ghost weight introduced by alinhac

global solution branches for a nonlocal allen cahn equation abstract we consider the neumann problem of a 1d stationary allen cahn equation with nonlocal term our previous paper 4 obtained a local branch of asymmetric solutions which bifurcates from a point on the branch of odd symmetric solutions this paper derives the global behavior of the branch of asymmetric solutions and moreover determines the set of all solutions to the nonlocal allen cahn equation our proof is based on a level set analysis for an integral map associated with the nonlocal term

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

blow up for quasi linear wave equations in three space dimensions we consider equations of the form u phi x t u u u 1a for a function u u x 1 x n t u x t here denotes the d alembertian square frac partial 2 partial t 2 vartriangle 1b and u represents the vector of first derivatives u that of second derivatives of u with respect to the x k and t on linearizing 1a shall go over into the classical wave equation u 0 this means that and its first derivatives with respect to u u u shall vanish for u u u 0

a note on discrete solutions of the plateau problem on demontre des theoremes pour la convergence de solutions discretes du probleme de plateau sous l hypothese que le contour de jordan soit rectifiable

front propagation problems with nonlocal terms ii abstract we study the propagation of hyper surfaces t of r n satisfying the equation v h x t where v is the normal velocity of t at x t is the interior of t and h is a given evolution law we prove the existence of generalized solutions and an inclusion principle for these generalized solutions copyright 2001 academic press dans cet article nous etudions certains problemes de propagation d hypersurfaces t de r n suivant des lois de la forme v h x t ou v est la vitesse normale a t en x t est l interieur de t et h est une loi d evolution on definit une notion de solutions generalisees et on montre l existence de telles solutions de plus on demontre un principe d inclusion pour ces solutions

generation motion and thickness of transition layers for a nonlocal allen cahn equation abstract we investigate the behavior as e 0 of the nonlocal allen cahn equation u t u 1 e 2 f u e u where f u 0 is of the bistable type given a rather general initial datum u 0 that is independent of e we perform a rigorous analysis of both the generation and the motion of the interface and obtain a new estimate for its thickness more precisely we show that the solution develops a steep transition layer within the time scale of order e 2 ln e and that the layer obeys the law of motion that coincides with the limit problem within an error margin of order e

an error estimate for a new scheme for mean curvature motion in this work we propose a new numerical scheme for the anisotropic mean curvature equation the solution of the scheme is not unique but for all numerical solutions we provide an error estimate between the continuous solution and the numerical approximation our scheme is also applicable to computing the solution to dislocation dynamics equations we also provide some numerical simulations

on almost global existence for nonrelativistic wave equations in 3d almost global solutions are constructed to three dimensional quadratically nonlinear wave equations the proof relies on generalized energy estimates and a new decay estimate the method applies to equations that are only classically invariant such as the nonlinear system of hyperelasticity 1996 john wiley sons inc

secondary bifurcation for a nonlocal allen cahn equation abstract this paper studies the neumann problem of a nonlocal allen cahn equation in an interval a main result finds a symmetry breaking secondary bifurcation point on the bifurcation curve of solutions with odd symmetry our proof is based on a level set analysis for the associated integral map a method using the complete elliptic integrals proves the uniqueness of secondary bifurcation point we also show some numerical simulations concerning the global bifurcation structure

a posteriori error estimates for the effective hamiltonian of dislocation dynamics we study an implicit and discontinuous scheme for a non local hamilton jacobi equation modelling dislocation dynamics for the evolution problem we prove an a posteriori estimate of crandall lions type for the error between continuous and discrete solutions we deduce an a posteriori error estimate for the effective hamiltonian associated to a stationary cell problem in dimension one and under suitable assumptions we also give improved a posteriori estimates numerical simulations are provided

a convergent monotone difference scheme for motion of level sets by mean curvature an explicit convergent finite difference scheme for motion of level sets by mean curvature is presented the scheme is defined on a cartesian grid using neighbors arranged approximately in a circle the accuracy of the scheme which depends on the radius of the circle dx and on the angular resolution d is formally o dx2 d the scheme is explicit and nonlinear the update involves computing the median of the values at the neighboring grid points numerical results suggest that despite the low accuracy acceptable results are achieved for small stencil sizes a numerical example is presented which shows that the centered difference scheme is non convergent

a new approach to front propagation problems theory and applications in this paper we present a new definition for the global in time propagation motion of fronts hypersurfaces boundaries with a prescribed normal velocity past the first time they develop singularities we show that if this propagation satisfies a geometric maximum principle inclusion avoidance type property then the normal velocity must depend only on the position of the front and its normal direction and principal curvatures this new approach which is more geometric and as it turns out equivalent to the level set method is then used to develop a very general and simple method to rigorously validate the appearance of moving interfaces at the asymptotic limit of general evolving systems like interacting particles and reaction diffusion equations we finally present a number of new asymptotic results among them are the asymptotics of i reaction diffusion equations with rapidly oscillating coefficients ii fully nonlinear nonlocal integral differential equations and iii stochastic ising models with long range anisotropic interactions and general spin flip dynamics

discrete solution of the plateau problem and its convergence definition d une solution discrete du probleme de plateau et demonstration des theoremes de convergence de la solution discrete lorsque la frontiere satisfait une certaine condition exemple numerique

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

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

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

layer oscillations in reaction diffusion systems this paper considers a two component system of reaction diffusion equations involving two parameters varepsilon and tau varepsilon tau u 1 varepsilon 2 u xx f u v qquad v

the generation and propagation of internal layers 4 c2u f u x 7 1 1 is studied where f is of bistable type for each x here bistable means that f vanishes for exactly three values of u and fu is negative at the two outer ones the prototype is f u u3 when f is independent of x the equation has a dynamical theory which is pretty well understood its most striking feature in that case is the existence of travelling wave solutions which are globally stable this is also true for the heterogeneous equation 1 l although such travelling wave solutions have up to now only been studied by formal asymptotics they are not travelling waves in the strictest sense but are rather frontal structures confined to thin zones of width o e which move with variable velocity starting with smooth arbitrary initial data us which do not depend on e the formation of these fronts at certain locations can be predicted and their approximate subsequent motion can be determined by solving an ordinary differential equation this heuristic analysis is explained in section 2 the purpose of this paper is to provide the heuristics with a rigorous footing in fact we obtain detailed estimates for the error made by approximating the exact solution by the function constructed in section 2 for simplicity we treat the case when there is only a single front although multiple fronts may clearly occur the process under consideration consists of two parts proceeding under two different time scales which are the initial generation of a layer followed by its propagation the analysis describing the first part is the more involved if comparison functions based on the ordinary differential equation in section 2 are used the analysis simplifies but weaker results are obtained

approximation schemes for propagation of fronts with nonlocal velocities and neumann boundary conditions the convergence of schemes for propagation of fronts in a bounded domain moving with normal velocities is studied the velocities considered depend on the principal curvatures the normal direction the location as well as some nonlocal properties of the front most of the schemes considered are in essence threshold dynamics type approximation schemes modified for neumann boundary conditions and nonlocal terms the existence and uniqueness of appropriately defined viscosity solutions of the level set equations describing the nonlocal motions is also shown

convolution generated motion as a link between cellular automata and continuum pattern dynamics cellular automata have been used to model the formation and dynamics of patterns in a variety of chemical biological and ecological systems however for patterns in which sharp interfaces form and propagate automata simulations can exhibit undesirable properties including spurious anisotropy and poor representation of interface curvature effects these simulations are also prohibitively slow when high accuracy is required even in two dimensions also the highly discrete nature of automata models makes theoretical analysis difficult in this paper we present a method for generating interface motions that is similar to the threshold dynamics type cellular automata but based on continuous convolutions rather than discrete sums these convolution generated motions naturally achieve the fine grid limit of the corresponding automata and they are also well suited to numerical and theoretical analysis because of this the desired pattern dynamics can be computed accurately and efficiently using adaptive resolution and fast fourier transform techniques and for a large class of convolutions the limiting interface motion laws can be derived analytically thus convolution generated motion provides a numerically and analytically tractable link between cellular automata models and the smooth features of pattern dynamics this is useful both as a means of describing the continuum limits of automata and as an independent foundation for expressing models for pattern dynamics in this latter role it also has a number of benefits over the traditional reaction diffusion ginzburg landau continuum pde models of pattern formation which yield true moving interfaces only as singular limits we illustrate the power of this approach with convolution generated motion models for pattern dynamics in developmental biology and excitable media

affine plane curve evolution a fully consistent scheme we present an accurate numerical scheme for the affine plane curve evolution and its morphological extension to grey level images this scheme is based on the iteration of a nonlocal fully affine invariant and numerically stable operator which can be exactly computed on polygons the properties of this operator ensure that a few iterations are sufficient to achieve a very good accuracy unlike classical finite difference schemes that generally require a lot of iterations convergence results are provided as well as theoretical examples and experiments

nonresonance and global existence of prestressed nonlinear elastic waves this article considers the existence of global classical solutions to the cauchy problem in nonlinear elastodynamics the unbounded elastic medium is assumed to be homogeneous isotropic and hyperelastic as in the theory of 3d nonlinear wave equations in three space dimensions global existence hinges on two basic assumptions first the initial deformation must be a small displacement from equilibrium in this case a prestressed homogeneous dilation of the reference conflguration and equally important the nonlinear terms must obey a type of nonresonance or null condition the omission of either of these assumptions can lead to the breakdown of solutions in flnite time in particular nonresonance complements the genuine nonlinearity condition of f john under which arbitrarily small spherically symmetric displacements develop singularities although one expects this to carry over to the nonsymmetric case as well 4 john also showed that small solutions exist almost globally 5 see also 10 formation of singularities for large displacements was illustrated by tahvildar zadeh 16 the nonresonance condition introduced here represents a substantial improvement over our previous work on this topic 13 to explain the difierence roughly our earlier version of the null condition forced the cancellation of all nonlinear wave interactions to flrst order along the characteristic cones here only the cancellation of nonlinear wave interactions among individual wave families is required the di culty in realizing this weaker version is that the decomposition of elastic waves into their longitudinal and transverse components involves the nonlocal helmholtz projection which is ill suited to nonlinear analysis however our decay estimates make clear that only the leading contribution of the resonant interactions along the characteristic cones is potentially dangerous and this permits the usage of approximate local decompositions

a real time algorithm for the approximation of level set based curve evolution in this paper we present a complete and practical algorithm for the approximation of level set based curve evolution suitable for real time implementation in particular we propose a two cycle algorithm to approximate level set based curve evolution without the need of solving partial differential equations pdes our algorithm is applicable to a broad class of evolution speeds that can be viewed as composed of a data dependent term and a curve smoothness regularization term we achieve curve evolution corresponding to such evolution speeds by separating the evolution process into two different cycles one cycle for the data dependent term and a second cycle for the smoothness regularization the smoothing term is derived from a gaussian filtering process in both cycles the evolution is realized through a simple element switching mechanism between two linked lists that implicitly represents the curve using an integer valued level set function by careful construction all the key evolution steps require only integer operations a consequence is that we obtain significant computation speedups compared to exact pde based approaches while obtaining excellent agreement with these methods for problems of practical engineering interest in particular the resulting algorithm is fast enough for use in real time video processing applications which we demonstrate through several image segmentation and video tracking experiments

almost global existence to nonlinear wave equations in three space dimensions on considere l existence a t grand de solutions de petite amplitude pour des equations d onde d ordre 2 non lineaires a 4 dimensions d espace temps

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

generation and propagation of interfaces in reaction diffusion systems this paper is concerned with the asymptotic behavior as e0 of the solution u e e of the second initial boundary value problem of the reaction diffusion system u t e e u e 1 e f u e e 1 e u e 1 u e2 e t e e u e e where 0 is a constant when 2 3 9 2 3 9 f is bistable in the sense that the ordinary differential equation u t f u has two stable solutions u h and u h and one unstable solution u h 0 where h h 0 and h are three solutions of the algebraic equation f u 0

convergence of nonlocal threshold dynamics approximations to front propagation in this note we prove that appropriately scaled threshold dynamics type algorithms corresponding to the fractional laplacian of order 0 2 converge to moving fronts when 1 the resulting interface moves by weighted mean curvature while for 1 the normal velocity is nonlocal of fractional type the results easily extend to general nonlocal anisotropic threshold dynamics schemes

threshold growth dynamics we study the asymptotic shape of the occupied region for monotone deterministic dynamics in d dimensional euclidean space parametrized by a threshold 0 and a borel set n r d with positive and finite lebesgue measure if a n denotes the oocupied set of the dynamics at integer time n then a n 1 is obtained by adjoining any point x for which the volume of overlap between x n and a n exceeds except in some degenerate cases we prove that n 1 a n converges to a unique limiting shape l starting from any bounded initial region a 0 that is suitably large moreover l is computed as the polar transform for 1 w where w is an explicit width function that depends on n and

numerical analysis of the model of image processing with time delay regularization in this paper we present a semi implicit numerical scheme for the model of image processing with time delay regularization and prove its convergence by the related theory of viscosity solution based on the principle of contraction mapping and discrete functional analysis we also establish the iterative convergence in the sense of l norm numerical experiments with gauss seidel iterative method of sparse matrix show that compared with the explicit scheme our numerical method make denoised image sharper

convolution kernels and stability of threshold dynamics methods threshold dynamics and its extensions have proven useful in computing interfacial motions in applications as diverse as materials science and machine learning certain desirable properties of the algorithm such as unconditional monotonicity in two phase flows and gradient stability more generally hinge on positivity properties of the convolution kernel and its fourier transform recent developments in the analysis of this class of algorithms indicate that sometimes as in the case of certain anisotropic curvature flows arising in materials science these properties of the convolution kernel cannot be expected other applications such as machine learning would benefit from as great a level of flexibility in choosing the convolution kernel as possible in this paper we establish certain desirable properties of threshold dynamics such as gamma convergence of its associated energy for a substantially wider class of kernels than has been hitherto possible we also present variants of the algorithm that

generation and propagation of interfaces for reaction diffusion equations abstract this paper is concerned with the asymptotic behavior as e 0 of the solution ue of the reaction diffusion equation in r n r u t 1 2 0 where is the derivative of a bistable potential we show that if the initial data u 0 has values in both domains of attraction of the potential then an interface will develop in a short time o e 2 ln e we also show that if the wells of the potential are of equal depth then this interface will propagate with normal velocity equal to its mean curvature our result is valid as long as the interface remains smooth if the initial interface is compact and n 2 then the interface will disappear in a finite time but not if n 1 in case the depths of the wells are not equal then in order to obtain get reasonable results we must work on the scaled time s t e slower time scale in this scale we show that the interface moves with a constant speed proportional to the difference of the depths of the two wells along the normal towards the domain of the deeper well this result is valid for all s 0 and does not actually depend on the regularity of the interface we also extend the above results to the homogeneous second initial boundary value problem in case the depths of the wells are equal and the initial interface is orthogonal to the boundary we prove that the interface moves with normal velocity equal to its mean curvature provided that there is a family of hypersurfaces which moves according to their mean curvature and intersect the boundary orthogonally

a mean field equation of motion for the dynamic ising model a mean field type of approximation is used to derive two differential equations one approximately representing the average behavior of the ising model with glauber spin flip stochastic dynamics and the other doing the same for kawasaki spin exchange dynamics the proposed new equations are compared with the cahn allen and cahn hilliard equations representing the same systems and with information about the exact behavior of the microscopic models

analysis of diffusion generated motion for mean curvature flow in codimension two a gradient flow approach the merriman bence osher mbo scheme also known as diffusion generated motion or thresholding is an efficient numerical algorithm for computing mean curvature flow mcf it is fairly well understood in the case of hypersurfaces this paper establishes the first convergence proof of the scheme in codimension two we concentrate on the case of the curvature motion of a filament curve in mathbb r 3 our proof is based on a new generalization of the minimizing movements interpretation for hypersurfaces esedoglu otto 15 by means of an energy that approximates the dirichlet energy of the state function as long as a smooth mcf exists we establish uniform energy estimates for the approximations away from the smooth solution and prove convergence towards this mcf the current result that holds in codimension two relies in a very crucial manner on a new sharp monotonicity formula for the thresholding energy this is an improvement of an earlier approximate version

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

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

diffusion generated motion of curves on surfaces we present a new method for computing the curvature driven motion of a curve constrained to move on a given surface it is based on the diffusion generated motion algorithm and retains both the novel simplicity of that method as well as the natural extension to curves with junctions to general geometric motion laws and to higher dimensions the result is an extremely simple algorithm for curvature dependent motion on surfaces wherein the only evolution operation is linear diffusion in three dimensional euclidean space

an algorithm for evolutionary surfaces an algorithm is presented which allows to split the calculation of the mean curvature flow of surfaces with or without boundary into a series of poisson problems on a series of surfaces this gives a new method to solve plateau s problem forh surfaces

on two methods for approximating minimal surfaces in parametric form etude de deux methodes d approximation des surfaces minimales sous forme parametrique minimisation de l aire et de l energie convergence transformation conforme approchee exemples

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

a simple proof of convergence for an approximation scheme for computing motions by mean curvature we prove the convergence of an approximation scheme recently proposed by bence merriman and osher for computing motions of hypersurfaces by mean curvature our proof is based on viscosity solutions methods

variational convergence of discrete minimal surfaces this work is concerned with the convergence behavior of the solutions to parametric variational problems an emphasis is put on sequences of variational problems that arise as discretizations of either infinite dimensional optimization problems or infinite dimensional operator problems finally the results are applied to discretizations of the douglas plateau problem and of a boundary value problem in nonlinear elasticity

nonlocal curvature flows this paper aims at building a unified framework to deal with a wide class of local and nonlocal translation invariant geometric flows we introduce a class of nonlocal generalized mean curvatures and prove the existence and uniqueness for the level set formulation of the corresponding geometric flows we then introduce a class of generalized perimeters whose first variation is an admissible generalized curvature within this class we implement a minimizing movements scheme and we prove that it approximates the viscosity solution of the corresponding level set pde we also describe several examples and applications besides recovering and presenting in a unified way existence uniqueness and approximation results for several geometric motions already studied and scattered in the literature the theory developed in this paper also allows us to establish new results

a convolution thresholding approximation of generalized curvature flows we construct a convolution thresholding approximation scheme for the geometric surface evolution in the case when the velocity of the surface at each point is a given function of the mean curvature conditions for the monotonicity of the scheme are found and the convergence of the approximations to the corresponding viscosity solution is proved we also discuss some aspects of the numerical implementation of such schemes and present several numerical results

existence of weak solutions for general nonlocal and nonlinear second order parabolic equations in this article we provide existence results for a general class of nonlocal and nonlinear second order parabolic equations the main motivation comes from front propagation theory in the cases when the normal velocity depends on the moving front in a nonlocal way among applications we present level set equations appearing in dislocation theory and in the study of fitzhugh nagumo systems

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

convergent difference schemes for nonlinear parabolic equations and mean curvature motion explicit finite difference schemes are given for a collection of parabolic equations which may have all of the following complex features degeneracy quasilinearity full nonlinearity and singularities in particular the equation of motion by mean curvature is included the schemes are monotone and consistent so that convergence is guaranteed by the general theory of approximation of viscosity solutions of fully nonlinear problems in addition an intriguing new type of nonlocal problem is analyzed which is related to the schemes and another very different sort of approximation is presented as well

consistency stability a priori and a posteriori errors for petrov galerkin methods applied to nonlinear problems in an abstract framework we present a formalism which specifies the notions of consistency and stability of petrov galerkin methods used to approximate nonlinear problems which are in many practical situations strongly nonlinear elliptic problems this formalism gives rise to a priori and a posteriori error estimates which can be used for the refinement of the mesh in adaptive finite element methods applied to elliptic nonlinear problems this theory is illustrated with the example div k u u c u f in a two dimensional domain with dirichlet boundary conditions

local existence and uniqueness of solutions of the stefan problem with surface tension and kinetic undercooling abstract we consider the stefan problem with surface tension and kinetic undercooling effects that is with the temperature u satisfying the condition u v on the interface const 0 where and v are the mean curvature and the normal velocity of respectively we establish short time existence and uniqueness of classical solutions for the resulting free boundary problem the key observation is that when translated to local coordinates equation is a quasi linear parabolic equation on a manifold without boundary to prove existence and uniqueness we look for the solution u as a fixed point of a contracting map r we start by solving the equation of motion for a given left hand side u this provides us with an interface which we use to solve the parabolic problem in the bulk with the conservation of energy condition across thereby obtaining a new temperature function r u finally it is the regularizing character of the operator r that allows us to show that r is a contraction on a small time interval

global solutions of nonlinear hyperbolic equations for small initial data on considere des systemes quasilineaires d equations hyperboliques d ordre 2 qui sont des deformations non lineaires de l equation d onde on etudie l existence globale et le comportement asymptotique des solutions du probleme de cauchy quand la donnee initiale est suffisamment petite

uniform decay estimates and the lorentz invariance of the classical wave equation on etudie le comportement asymptotique des solutions de u 0 ou t 2 1 2 n 2 pour des conditions initiales u 0 u t g x en t 0 avec g reguliere a support compact dans r n

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

convergence of a generalized fast marching method for an eikonal equation with a velocity changing sign we present a new fast marching algorithm for an eikonal equation with a velocity changing sign this first order equation models a front propagation in the normal direction the algorithm is an extension of the fast marching method in two respects the first is that the new scheme can deal with a time dependent velocity and the second is that there is no restriction on its change in sign we analyze the properties of the algorithm and we prove its convergence in the class of discontinuous viscosity solutions finally we present some numerical simulations of fronts propagating in mathbb r 2

convergence of a generalized fast marching method for a non convex eikonal equation we present a new fast marching algorithm for a non convex eikonal equation modeling front evolutions in the normal direction the algorithm is an extension of the fast marching method since the new scheme can deal with a emph time dependent velocity without emph any restriction on its sign we analyze the properties of the algorithm and we prove its convergence in the class of discontinuous viscosity solutions finally we present some numerical simulations of fronts propagating in r 2

convergence of approximation schemes for nonlocal front propagation equations we provide a convergence result for numerical schemes approximating nonlocal front propagation equations our schemes are based on a recently investigated notion of a weak solution for these equations we also give examples of such schemes for a dislocation dynamics equation and for a fitzhugh nagumo type system

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

multiclass data segmentation using diffuse interface methods on graphs we present two graph based algorithms for multiclass segmentation of high dimensional data on graphs the algorithms use a diffuse interface model based on the ginzburg landau functional related to total variation and graph cuts a multiclass extension is introduced using the gibbs simplex with the functional s double well potential modified to handle the multiclass case the first algorithm minimizes the functional using a convex splitting numerical scheme the second algorithm uses a graph adaptation of the classical numerical merriman bence osher mbo scheme which alternates between diffusion and thresholding we demonstrate the performance of both algorithms experimentally on synthetic data image labeling and several benchmark data sets such as mnist coil and webkb we also make use of fast numerical solvers for finding the eigenvectors and eigenvalues of the graph laplacian and take advantage of the sparsity of the matrix experiments indicate that the results are competitive with or better than the current state of the art in multiclass graph based segmentation algorithms for high dimensional data

on the numerical approximation of unstableminimal surfaces with polygonal boundaries this work is concerned with the approximation and the numerical computation of polygonal minimal surfaces in bbb r q q ge 2 polygonal minimal surfaces correspond to the critical points of shiffman s function theta since this function is analytic polygonal minimal surfaces can be characterized by means of the second derivative of theta we present a finite element approximation of quasiminimal surfaces together with an error estimate in this way we obtain discrete approximations theta h of theta and f h of nabla theta in particular we prove that the discrete functions converge uniformly on certain compact subsets this will be the main tool for proving existence and convergence of discrete minimal surfaces in neighbourhoods of non degenerate minimal surfaces in the numerical part of this paper we compute numerical approximations of polygonal minimal surfaces by use of newton s method applied to f h