slow motion manifolds far from the attractor in multistable reaction diffusion equations abstract we consider a scalar reaction diffusion equation with multistable nonlinearity with a particular symmetry by reduction to a family of transmission problems in r and by contraction arguments a manifold close to an invariant manifold formed by functions exhibiting a pattern of transition layers is constructed an approximation for the associated vector field is also provided this shows that the motion on those manifolds is exponentially slow as in the well known case of the bistable equation however in opposition to the bistable case some of these manifolds are far from the attractor since these manifolds correspond to metastable patterns this shows the importance of the transient motion toward the attractor and the importance of these manifolds in organizing that motion it is also shown that by a suitable perturbation we can obtain new equilibria on those manifolds

closed solution for the spatially homogeneous kac s model of the nonlinear boltzmann equation finds a closed spatially homogeneous solution of the nonlinear kac s model 1956 in 1 1 dimensions velocity v and time t choosing for the even velocity part of the distribution function the bobylev krook wu mode 1975 6 the author adds an odd velocity part the author finds the possibility of the existence of the tjon relaxation effect when the time t is increasing this depends on both the initial condition and the cross section

relativistic gamow vectors the friedrichs model has often been used in order to obtain explicit formulas for eigenvectors associated to complex eigenvalues corresponding to lifetimes such eigenvectors are called gamow vectors and they acquire meaning in extensions of the conventional hilbert space of quantum theory to the so called rigged hilbert space in this paper gamow vectors are constructed for a solvable model of an unstable relativistic field as a result we obtain a time asymmetric relativistic extension of the fock space this extension leads to two distinct poincare semigroups the time reversal transformation maps one semigroup to the other as a result the usual pct invariance should be extended we show that irreversibility as expressed by dynamical semigroups is compatible with the requirements of relativity

dynamics of one and two dimensional fronts in a bistable equation with time delayed global feedback propagation failure and control mechanisms we study the evolution of fronts in a bistable equation with time delayed global feedback in the fast reaction and slow diffusion regime this equation generalizes the hodgkin grafstein and allen cahn equations we derive a nonlinear equation governing the motion of fronts which includes a term with delay in the one dimensional case this equation is linear we study the motion of one and two dimensional fronts finding a much richer dynamics than for the previously studied cases without time delayed global feedback we explain the mechanism by which localized fronts created by inhibitory global coupling loose stability in a hopf bifurcation as the delay time increases we show that for certain delay times the prevailing phase is different from that corresponding to the system in the absence of global coupling numerical simulations of the partial differential equation are in agreement with the analytical predictions

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

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

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

an exact approach to the oscillator radiation process in an arbitrarily large cavity starting from a solution of the problem of a mechanical oscillator coupled to a scalar field inside a reflecting sphere of radius r we study the behaviour of the system in free space as the limit of an arbitrarily large radius in the confined solution from a mathematical point of view we show that this way of addressing the problem is not equivalent to considering the system a priori embedded in infinite space in particular the matrix elements of the transformation turning the system to the principal axis do not tend to distributions in the limit of an arbitrarily large sphere as should be the case if the two procedures were mathematically equivalent also we introduce dressed coordinates which allow an exact description of the oscillator radiation process expanding in powers of the coupling constant we recover from our exact expressions the well known decay formulae from perturbation theory

gradient flow and front propagation with boundary contact energy we consider a gradient flow for the functional f epsilon u int omega frac 1 epsilon w u epsilon nabla u 2 dx int partial omega sigma u d h n 1 x epsilon 0 with a view towards modelling the interfacial motion associated with domain coarsening in binary alloys in the presence of boundary contact energy here w is a non negative double well free energy density and u is the composition density of the system the function sigma represents the contact energy between the alloy and the boundary partial omega of the container the functional was first proposed by cahn in a different context as a model for the energy of a two fluid mixture which takes into account boundary contact energy at the boundary of the fluid domain we derive the first term in an asymptotic expansion as epsilon 0 for the solution of the associated gradient flow begin eqnarray partial t u 2 epsilon delta u 1 epsilon w u x in omega t geq 0 u x 0 g epsilon x x in omega nabla u cdot bf n 1 2 epsilon sigma u x in partial omega t geq 0 end eqnarray using multiple time scales we show that fronts rapidly develop and then propagate with normal velocity epsilon kappa where kappa is the mean curvature of the front at the intersection of a front with partial omega the boundary contact energy is shown to imply a contact angle condition for the front several examples are presented for this type of propagation in the plane

geometric optics approach to reaction diffusion equations the speed of propagation of traveling waves of reaction diffusion equations is calculated by an asymptotic method similar to geometrical optics this also works for reaction diffusion waves in random media

motion by curvature by scaling nonlocal evolution equations we prove convergence to a motion by mean curvature by scaling diffusively a nonlinear nonlocal evolution equation this equation was introduced earlier to describe the macroscopic behavior of a ferromagnetic spin system with kac interaction which evolves with glauber dynamics the convergence is proven in any time interval in which the limiting motion is regular

nonlinear kac model spatially homogeneous solutions and the tjon effect we study kac s nonlinear model of the boltzmann equation when the cross section does not satisfy the special symmetry condition we determine a differential system for the laguerre moments of the odd and even velocity parts of the solutions we consider the spatially homogeneous model in 1 1 dimensions velocityv and timet when the even velocity part of the solution is provided by the bobylev krook wu closed solutions and study the associated odd velocity part we find that the solutions depend on the microscopic models of for one class of which has sums of exponential terms for the laguerre moments we establish the relations allowing the construction of the time dependent solutions associated with any initial distribution we find sufficient conditions on and on the even part such that the laguerre series of the odd part converges we establish a criterion for a well defined linear combination of the moments cross section and we check its validity for different numerical examples we find that if the relaxation time for the even part is smaller than the corresponding one for the odd part and if the initial distribution has a narrow peak then the tjon effect exists for the complete b k w solution even odd parts

nonparabolic dissipative systems modeling the flow of liquid crystals we study a simplified system which retains most of the interesting mathematical properties of the original ericksen leslie equations for the flow of liquid crystals this is a coupled nonparabolic dissipative dynamic system we derive several energy laws which enable us to prove the global existence of the weak solutions and the classical solutions we also discuss uniqueness and some stability properties of the system 1995 john wiley sons inc

exact solution for the spatially homogeneous nonlinear kac model of the boltzmann equation with an external force term the author studies the spatially homogeneous kac model of the nonlinear boltzmann equation in 1 1 dimensions velocity v and time t when an external force term is present the author finds a closed solution the force term decreases exponentially with time like constant1 exp constant2 t where both constants depend explicitly on the moments of the cross section the author finds for the relaxation to equilibrium that the tjon overpopulation effect depends on both the initial condition and the microscopic model of cross section for the existence of this effect the author establishes a criterion which is a well defined linear combination of the moments of the author cross section

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

on the slowness of phase boundary motion in one space dimension we study the limiting behavior of the solution of u t e 2 u xx u 3 u 0 a x b with a neumann boundary condition or an appropriate dirichlet condition the analysis is based on energy methods we assume that the initial data has a transition layer structure i e ue 1 except near finitely many transition points we show that in the limit as e 0 the solution maintains its transition layer structure and the transition points move slower than any power of e

closed similarity solutions for a class of stationary nonlinear boltzmann like equations the authors look for closed stationary solutions of nonlinear transport equations of the stochastic boltzmann type in 1 1 dimensions velocity v and position x these solutions mainly written as the product of an exponential by a polynomial in z xlambda v2 have the quasi similarity property as a consequence of their polynomial character they violate the positivity requirement however the authors think that they are the dominant term of infinite series solutions which might have the desired positivity property calculations are greatly simplified by using sum rules analogous to conservation of mass and energy

semi inhomogeneous solutions of the kac model of boltzmann equations constructs semi inhomogeneous solutions of the kac model for which the spatial dependence is only present in the odd velocity part of the distribution at the macroscopic level the local density the local energy and the associated current components are determined and discussed while at the microscopic level there exist two different classes of distributions as an illustration the author gives two solutions explicitly for one class the distributions relax towards a maxwellian equilibrium solution and they correspond either to a contraction or an expansion while for the other they go to zero when the time increases to infinity the space variable appears linearly and in order to maintain the positivity stays inside a finite interval assuming that the distributions are zero outside such intervals a physical interpretation can be obtained for the whole space axis provided appropriate elastic walls sinks and sources are introduced these boundary conditions seem more natural for the class of distributions with maxwellian relaxation

existence and regularity of mean curvature flow with transport term in higher dimensions given an initial c 1 hypersurface and a time dependent vector field in a sobolev space we prove a time global existence of a family of hypersurfaces which start from the given hypersurface and which move by the velocity equal to the mean curvature plus the given vector field we show that the hypersurfaces are c 1 for a short time and even after some singularities occur almost everywhere c 1 away from the higher multiplicity region

a discrete model for an ill posed nonlinear parabolic pde we study a finite difference discretization of an ill posed nonlinear parabolic partial differential equation the pde is the one dimensional version of a simplified two dimensional model for the formation of shear bands via anti plane shear of a granular medium for the discretized initial value problem we derive analytically and observed numerically a two stage evolution leading to a steady state i an initial growth of grid scale instabilities and ii coarsening dynamics elaborating the second phase at any fixed time the solution has a piecewise linear profile with a finite number of shear bands in this coarsening phase one shear band after another collapses until a steady state with just one jump discontinuity is achieved the amplitude of this steady state shear band is derived analytically but due to the ill posedness of the underlying problem its position exhibits sensitive dependence analyzing data from the simulations we observe that the number of shear bands at time t decays like t 1 3 from this scaling law we show that the time scale of the coarsening phase in the evolution of this model for granular media critically depends on the discreteness of the model our analysis also has implications to related ill posed nonlinear pdes for the one dimensional perona malik equation in image processing and to models for clustering instabilities in granular materials 2001 elsevier science b v all rights reserved

the singular limit of the allen cahn equation and the fitzhugh nagumo system abstract we consider an allen cahn type equation of the form u t u e 2 f e x t u where e is a small parameter and f e x t u f u e g e x t u a bistable nonlinearity associated with a double well potential whose well depths can be slightly unbalanced given a rather general initial data u 0 that is independent of e we perform a rigorous analysis of both the generation and the motion of interface 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 formal asymptotic limit within an error margin of order e this is an optimal estimate that has not been known before for solutions with general initial data even in the case where g e 0 next we consider systems of reaction diffusion equations of the form u t u e 2 f e u v v t d v h u v which include the fitzhugh nagumo system as a special case given a rather general initial data u 0 v 0 we show that the component u develops a steep transition layer and that all the above mentioned results remain true for the u component of these systems

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

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

the effects of obstacle size on periodic travelling waves in oscillatory reaction diffusion equations many natural populations undergo multi year cycles and field studies have shown that these can be organized into periodic travelling waves ptws mathematical studies have shown that large scale landscape obstacles represent a natural mechanism for wave generation here we investigate how the amplitude and wavelength of the selected waves depend on the obstacle size we firstly consider a large circular obstacle in an infinite domain for a reaction diffusion system of l u type we use perturbation theory to derive a leading order approximation to the wave generated by the obstacle this shows the dependence of the wave properties on both parameter values and obstacle size we find that the limiting values of the amplitude and wavelength are approached algebraically with distance from the obstacle edge rather than exponentially in the case of a flat boundary we use our results to predict the properties of waves generated by a large circular obstacle for an oscillatory predator prey system via a reduction of the predator prey model to normal form close to hopf bifurcation our predictions compare well with numerical simulations we also discuss the implications of these results for wave stability and briefly investigate the effects of obstacles with elliptical geometries

the location and stability of interface solutions of an inhomogeneous parabolic problem reaction diffusion mechanisms have been used to explain pattern formation in developmental biology and in experimental chemical systems environmental effects are included through an inhomogeneous nonlinear forcing term in a parabolic equation model with a small diffusion coefficient the small diffusivity and the form of the nonlinear forcing term yield interface solutions after a time of order one and these interface solutions then change slowly on a longer timescale the stability of these interface solutions is considered from both a numerical viewpoint and using asymptotic analysis and it is found that a simple sign condition determines stability here the existence of a steady state interface is determined by the zeroes of the derivative of a function h x equiv ln f x g3 x and the stability of this steady state interface which determines a patterned solution is controlled by the sign of the second derivative of h x at a zero where the interface occurs

pattern formation capacity of spatially extended systems we analyze a class of spatially extended systems which are capable of generating many complicated patterns these systems are given by the ginzburg landau equation coupled with a system of two linear equations and describe nonlinear media with localized defects we find a connection between these systems and spin glass systems we show that the system is capable to produce many patterns and describe patterning algorithms

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

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

development of interfaces in ℝ n consider the reaction diffusion equation in n u t h 2 u u 0 where is the derivative of a bistable even potential and h is a small parameter if the initial data have a smooth noncritical zero set we prove that an interface appears in time o log h 1 and that the solution stays close to it for at least time o 1 h

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

coarsening of step bunches in step flow growth a reaction diffusion model and its travelling wave solutions abstract we consider a model of the frank impurity mechanism in step flow growth based on reaction diffusion equations the equations based on the bcf model of crystal growth are designed to capture the physics of both the diffusion of adatoms and the appearance of impurities on the crystal surface following the model proposed by frank the model was originally considered by kandel and weeks physica d 66 1993 78 phys rev b 49 1994 5554 phys rev b 52 1995 2154 the model is a fundamentally two dimensional one it exhibits quick formation of step bunches followed by a much longer period of coarsening in the direction lateral to the step flow direction using tools from the theory of reaction diffusion equations we are able to reduce the complicated pattern formation of the model to a simple dynamical picture in most regions step bunches form quickly and these step bunches are equilibrium solutions of the equations as the step bunches form transitions form between different step bunches these transitions are travelling wave solutions of the equations these travelling waves govern the coarsening process the spatial patterns coarsen as the different travelling waves march across the surface of the crystal and encounter and annihilate each other by classifying the equilibrium states and travelling waves of the equations we can collapse the number of travelling waves to a small reduced set this reduction of the dynamics to travelling waves from a small set of equivalent classes is the basis of a simple reduced model which despite its simplicity can capture in entirety the coarsening process arising from the original equations using this reduced model we can investigate large scale nature of the coarsening process and behaviour of the coarsening process and reconsider some of the issues raised by kandel and weeks in their original analysis

image segmentation using some piecewise constant level set methods with mbo type of projection in this work we are trying to propose fast algorithms for mumford shah image segmentation using some recently proposed piecewise constant level set methods pclsm two variants of the pclsm will be considered in this work the first variant which we call the binary level set method needs a level set function which only takes values 1 to identify the regions the second variant only needs to use one piecewise constant level set function to identify arbitrary number of regions for the mumford shah image segmentation model with these new level set methods one needs to minimize some smooth energy functionals under some constrains a penalty method will be used to deal with the constraint aos additive operator splitting and mos multiplicative operator splitting schemes will be used to solve the euler lagrange equations for the minimization problems by doing this we obtain some algorithms which are essentially applying the mbo scheme for our segmentation models advantages and disadvantages are discussed for the proposed schemes

spiral wave solutions for reaction diffusion equations in a fast reaction slow diffusion limit abstract a two species reaction diffusion equation is studied in a distinguished limit of fast reaction and slow diffusion asymptotic matching yields a spiral solution whose rotation frequency is a unique function of the physical parameters and the constant pitch of the spiral using an idea of fife the problem is first reduced to a free boundary problem the solution is obtained by matching the core where diffusion balances the curvature of the free boundary to the far field where the effects of diffusion are small the equations of motion are averaged over the rotation frequency to yield steady equations in the two regions helically symmetric solutions to the averaged equation reduce to solving a fourth order ode in the core region matched to a second order ode in the far field speculations on the dynamics of the system and coupling between the core and far field are presented

reinitialization free level set evolution via reaction diffusion this paper presents a novel reaction diffusion rd method for implicit active contours that is completely free of the costly reinitialization procedure in level set evolution lse a diffusion term is introduced into lse resulting in an rd lse equation from which a piecewise constant solution can be derived in order to obtain a stable numerical solution from the rd based lse we propose a two step splitting method to iteratively solve the rd lse equation where we first iterate the lse equation then solve the diffusion equation the second step regularizes the level set function obtained in the first step to ensure stability and thus the complex and costly reinitialization procedure is completely eliminated from lse by successfully applying diffusion to lse the rd lse model is stable by means of the simple finite difference method which is very easy to implement the proposed rd method can be generalized to solve the lse for both variational level set method and partial differential equation based level set method the rd lse method shows very good performance on boundary antileakage the extensive and promising experimental results on synthetic and real images validate the effectiveness of the proposed rd lse approach

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

the effect of a singular perturbation on nonconvex variational problems we study the effect of a singular perturbation on certain nonconvex variational problems the goal is to characterize the limit of minimizers as some perturbation parameter 0 the technique utilizes the notion of convergence of variational problems developed by de giorgi the essential idea is to identify the first nontrivial term in an asymptotic expansion for the energy of the perturbed problem in so doing one characterizes the limit of minimizers as the solution of a new variational problem for the cases considered here these new problems have a simple geometric nature involving minimal surfaces and geodesics

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

metastable patterns in solutions of ut ϵ2uxx f u we consider the equation in question on the interval 0 x 1 having neumann boundary conditions with f u f u where f is a double well energy density with equal minima at u 1 the only stable states of the system are patternless constant solutions but given two phase initial data a pattern of interfacial layers typically forms far out of equilibrium the ensuing nonlinear relaxation process is extremely slow patterns persist for exponentially long times proportional to exp a l where a f 1 and l is the minimum distance between layers physically a tiny potential jump across a layer drives its motion we prove the existence and persistence of these metastable patterns and characterise accurately the equations governing their motion the point of view is reminiscent of center manifold theory a manifold parametrising slowly evolving states is introduced a neighbourhood is shown to be normally attracting and the parallel flow is characterised to high relative accuracy proofs involve a detailed study of the dirichlet problem spectral gap analysis and energy estimates

the gradient theory of phase transitions for systems with two potential wells on generalise la theorie du gradient des transitions de phase au cas a valeur vectoriel on considere la famille des perturbations e e u w u dx e 2 u 2 dx de la fonctionnelle non convexe e 0 u w u dx ou w r n r supporte deux phases et n 1

motion by curvature of planar curves with end points moving freely on a line this paper deals with the motion by curvature of planar curves having end points moving freely along a line with fixed contact angles to this line we first prove the existence and uniqueness of self similar shrinking solution then we show that the curve shrinks to a point in a self similar manner if initially the curve is a graph

dressed coordinates the path integral approach the recently introduced dressed coordinates are studied in the path integral approach these coordinates are defined in the context of a harmonic oscillator linearly coupled to massless scalar field and it is shown that in this model the dressed coordinates appear as a coordinate transformation preserving the path integral functional measure the analysis also generalizes the sum rules established in a previous work

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

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

motion of vortex lines in the ginzburg landau model equations of motion for vortex lines in the ginzburg landau theory are derived we construct asymptotic approximations for the complex order parameter which are valid in the core region and in the far field using the method of matched asymptotic expansions we show that to leading order the line moves in the binormal direction with a curvature dependent velocity we also consider the contribution of remote parts of the line interaction between several vortex lines and interaction with external fields

phase transition for potentials of high dimensional wells for a potential function f w r k rc that attains its global minimum value at two disjoint compact connected submanifolds n in r k we discuss the asymptotics as 0 of minimizers u of the singular perturbed functional e u d r jruj 2 c 1 2 f u dx under suitable dirichlet boundary data g w r k in the expansion of e u with respect to 1 we identify the first order term by the area of the sharp interface between the two phases an area minimizing hypersurfacea and the energyc f of minimal connecting orbits between n c and n and the zerothorder term by the energy of minimizing harmonic maps into n both under the dirichlet boundary condition on and a very interesting partially constrained boundary condition on the sharp interfacea 2011 wiley periodicals inc

exact solution for the spatially inhomogeneous nonlinear kac model of the boltzmann equation we study the spatially inhomogeneous kac model of the nonlinear boltzmann equation in 1 1 1 dimensions velocity v time t position x we obtain an exact solution which is the product of a maxwellian with a time dependent width by a second order polynomial in the velocity variable the solution satisfies a specular reflection condition at the boundary x x0 the position x0 x appears linearly and only in the odd part of the velocity distribution the range of x0 x being arbitrarily large but finite in order to maintain the positivity of the distribution the local density is spatially homogeneous further a particular linear relation between the moments of the cross section must be satisfied the most general maxwellian width has two relaxation times and their ratio is a function of the moments of the cross sections depending on whether this ratio is larger than or smaller than 1 we find contraction or expansion the solution relaxes towards a maxwellian equilibrium solution studying the tjon overpopu

front interaction and nonhomogeneous equilibria for tristable reaction diffusion equations the authors investigate the interfacial dynamics associated with three phase systems and the possible occurrence of steady multiphase patterns in one and higher space dimensions the model for this study is a ginzburg landau model for phase transitions given by the partial differential equation u 1 varepsilon delta u frac 1 varepsilon v u varepsilon u defined for x in a domain omega subset r and t 0 the potential v varepsilon is assumed to possess three wells having depths within mathcal o varepsilon of each other the evolution of the solution can be described by tracking the motion of fronts connecting adjacent wells of v varepsilon the primary interest is to understand how the self induced motion of a given front is affected by the presence of a second nearby front this interaction is characterized precisely using the formal method of matched asymptotic expansions in some settings this leads to the prediction of steady waves linking the outer two phases

slow motion for gradient systems with equal depth multiple well potentials abstract for scalar reaction diffusion in one space dimension it has been known for a long time that fronts move with an exponentially small speed for potentials with several distinct minimizers the purpose of this paper is to provide a similar result in the case of systems our method relies on a careful study of the evolution of localized energy this approach also has the advantage of relaxing the preparedness assumptions on the initial datum

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

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

a singularly perturbed reaction diffusion problem with inhomogeneous environment reaction diffusion problems have been used to describe pattern formation in developmental biology and material science i study the existence and stability of a singularly perturbed reaction diffusion problem with inhomogeneous environment in one dimensional space domain and also high dimensional space domain

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