li yau gradient bounds on compact manifolds under nearly optimal curvature conditions abstract we prove li yau type gradient bounds for the heat equation either on manifolds with fixed metric or under the ricci flow in the former case the curvature condition is r i c l p for some p n 2 or sup m m r i c 2 y d 2 n x y d y where n is the dimension of the manifold in the later case one only needs scalar curvature being bounded we will explain why the conditions are nearly optimal and give an application the li yau bound for the heat equation on manifolds with fixed metric seems to be the first one allowing ricci curvature not bounded from below

diffuse interface modeling of two phase flows based on averaging mass and momentum equations a diffuse interface model is derived for the direct simulation of two phase flows with surface tension phase change and density and viscosity differences between the phases the derivation starts from the balance equations for a sharp interface and uses an ensemble averaging procedure on an atomic scale to obtain a diffuse interface version of the equations as opposed to thermodynamically derived models the two phases are assumed to coexist inside the diffuse interface with different properties velocities and pressures separate conservation equations are solved for each phase the phase interactions are modeled explicitly through the inclusion of interfacial forces in the momentum equations for each phase based on a superposition of microscopic atomic scale and macroscopic interface morphologies an expression for the interfacial momentum source due to surface tension is introduced that is equivalent to the capillary stress term encountered in thermodynamically derived models also a constitutive relation for the average viscous stresses of each phase inside the diffuse interface is presented the model is tested for simple one dimensional flows tangential and normal to a diffuse interface and the results are compared to those obtained from a thermodynamically derived model

on the heat kernel of the bergmann metric on algebraic varieties the restriction of the standard fubini study metric of ycpn l to m m is an incomplete kahler metric g called the bergmann metric let us consider the operators d and 3 defined on c1 functions and c1 1 forms on m m respectively we define the domain 9y d of d to be the set of c1 functions 2 f defined on m m such that both f and df are in l similarly we define the domain 92 6 of to be the set of c1 1 forms co such that both co and 2 6 o are in l we then define the laplacian a with respect to the bergmann metric by a d with domain 9 a given by the set of c2 functions f such that f e 9 d and df e 6 the main purpose of this paper is to show that

remarks on differential harnack inequalities abstract in this paper we give a generalization of global and local differential harnack inequalities for heat equations obtained by li and xu j f li x j xu differential harnack inequalities on riemannian manifolds i linear heat equation adv math 226 5 2011 4456 4491 and baudoin and garofalo f baudoin n garofalo perelman s entropy and doubling property on riemannian manifolds j geom anal 21 2011 1119 1131 from this we can derive new harnack inequalities and new lower bounds for the associated heat kernel also we provide some new entropy formulas with monotonicity

numerical testing of quantitative phase field models with different polynomials for isothermal solidification in binary alloys quantitative phase field models have been developed as feasible computational tools for solving the free boundary problem in solidification processes these models are constructed with some polynomials of the phase field variable that describe variations of the physical quantities inside the diffuse interface the accuracy of the simulation depends on the choice of the polynomials and such dependence is indispensable for high performance computing and valuable for extending the range of applications of the model to several physical systems however little is known about the dependence of the accuracy on the choice of the polynomials in this study numerical testing is carried out for quantitative phase field models with extensive sets of polynomials 24 different models for isothermal solidification in binary alloys it is demonstrated in two dimensional simulations of dendritic growth that a specific set of polynomials must be employed to achieve high accuracy in the models with double well and double obstacle potentials both types of model with the best set of polynomials yield almost the same numerical accuracy

li yau and harnack type inequalities in rcd k n metric measure spaces abstract metric measure spaces satisfying the reduced curvature dimension condition cd k n and where the heat flow is linear are called rcd k n spaces this class of non smooth spaces contains gromov hausdorff limits of riemannian manifolds with ricci curvature bounded below by k and dimension bounded above by n we prove that in rcd k n spaces the following properties of the heat flow hold true a li yau type inequality a bakry qian inequality the harnack inequality

the swift hohenberg equation on conic manifolds abstract we consider the swift hohenberg equation on manifolds with conical singularities and show existence uniqueness and maximal regularity of the short time solution in terms of mellin sobolev spaces moreover we give a necessary and sufficient condition so that the above solution exists for all times space asymptotic expansion of the solution near the singularity is also provided and its relation to the local geometry is shown the same problem is considered on closed manifolds and similar results are obtained by using the above singular analysis theory

local elliptic gradient estimates for a nonlinear parabolic equation under the ricci flow abstract in this study we derive local elliptic hamilton type and souplet zhang type gradient estimates for positive solutions of the nonlinear parabolic partial differential equation t t u q u a u ln u under the ricci flow

hamilton s gradient estimates for fast diffusion equations under the ricci flow abstract let m be a complete noncompact riemannian manifold of dimension n we derive a hamilton s gradient estimate for positive solutions of the fast diffusion equations u t u m 1 4 n 8 m 1 on m 0 under the ricci flow

elliptic gradient estimates for a nonlinear heat equation and applications abstract in this paper we study elliptic gradient estimates for a nonlinear f heat equation which is related to the gradient ricci soliton and the weighted log sobolev constant of smooth metric measure spaces precisely we obtain hamilton s and souplet zhang s gradient estimates for positive solutions to the nonlinear f heat equation only assuming the bakry emery ricci tensor is bounded below as applications we prove parabolic liouville properties for some kind of ancient solutions to the nonlinear f heat equation some special cases are also discussed

on the equivalence of the entropic curvature dimension condition and bochner s inequality on metric measure spaces we prove the equivalence of the curvature dimension bounds of lott sturm villani via entropy and optimal transport and of bakry emery via energy and gamma 2 calculus in complete generality for infinitesimally hilbertian metric measure spaces in particular we establish the full bochner inequality on such metric measure spaces moreover we deduce new contraction bounds for the heat flow on riemannian manifolds and on mms in terms of the l 2 wasserstein distance

gradient estimates and harnack inequalities of a parabolic equation under geometric flow abstract in this paper we consider a manifold evolving by a general geometric flow and study parabolic equation q x t t u x t a u x t x t m 0 t we establish space time gradient estimates for positive solutions and elliptic type gradient estimates for bounded positive solutions of this equation by integrating the gradient estimates we derive the corresponding harnack inequalities finally as applications we give gradient estimates of some specific parabolic equations

gradient estimates and liouville theorems for nonlinear parabolic equations on noncompact riemannian manifolds abstract let m be a complete noncompact riemannian manifold in this paper we derive a local gradient estimate for positive solutions of the nonlinear parabolic equation t u x t x t u x t 0 on m 0 we also obtain a theorem of liouville type for positive solutions of this nonlinear equation this paper extends the result of souplet and zhang p souplet qi s zhang sharp gradient estimate and yau s liouville theorem for the heat equation on noncompact manifolds bull lond math soc 38 2006 1045 1053

on the upper estimate of the heat kernel of a complete riemannian manifold let m be a complete non compact riemannian manifold whose sectional curvature is bounded between two constants k and k then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in euclidean space the purpose of this paper is to give a justification of such a statement in 5 j cheeger and the third author found a lower estimate of the heat kernel by comparing it with the heat kernel of the space form whose curvature is the lower bound of the curvature of the manifold this lower estimate is sharp if we insist the dependence should be on the lower bound of the ricci curvature alone it remains to give an upper estimate of the heat kernel one does not however expect to have a comparison theorem for the upper bound because it is more sensitive to the geometry of the manifold in fact the heat kernel of the upper hialf space and the heat kernel of the complete hyperbolic manifold with finite volume have quite different behavior this is reflected by the fact that the laplacian has no discrete spectrum in the first case while infinite number of discrete eigenvalues may exist in the latter case what we will prove here is that in any case the heat kernel has to decay in a manner similar to the euclidean heat kernel thus we will prove that for any constant c 4 there exists c1 depending on c t the bound of the curvature of m and x so that for all t e 0 t

gaussian upper bounds for the heat kernels of some second order operators on riemannian manifolds abstract we describe a method of obtaining gaussian upper bounds on heat kernels which unifies and improves recent results for hypoelliptic operators in divergence form and for laplace beltrami operators on complete riemannian manifolds

large time behavior of schrodinger heat kernels and applications we obtain global in time and qualitatively sharp bounds for the heat kernel g of the schrodinger operator v the potential v satisfies v x c d x b near infinity with b 0 2 when v 0 our result can be described as follows g is bounded from above and below by the multiples of standard gaussians with a weight function if b 2 then the weight is bounded between two positive it constants if b 2 the weight is bounded between two positive functions of t d x and d y which have polynomial decay if b 2 or b 0 an application to a semilinear elliptic problem is also given

superlinear elliptic inequalities on manifolds abstract let m be a complete non compact riemannian manifold and let be a radon measure on m we study the problem of existence or non existence of positive solutions to a semilinear elliptic inequality u u q in m where q 1 we obtain necessary and sufficient criteria for existence of positive solutions in terms of green function of in particular explicit necessary and sufficient conditions are given when m has nonnegative ricci curvature everywhere in m or more generally when green s function satisfies the 3g inequality

essential spectrum and heat kernel abstract upper bounds are derived for the large time behavior of the heat kernel on riemannian manifolds whose essential spectrum has a positive lower bound applications include a relative signature theorem and the construction of bounded harmonic functions

a cameron martin type quasi invariance theorem for pinned brownian motion on a compact riemannian manifold the results in driver 13 for quasi invariance of wiener measure on the path space of a compact riemannian manifold m are extended to the case of pinned wiener measure to be more explicit let h 0 1 tom be a c1 function where m is a compact riemannian manifold o e m is a base point and tom is the tangent space to m at o e m let w m be the space of continuous paths from 0 1 into m v be wiener measure on w m concentrated on paths starting at o e m and hs w denote the stochastic parallel translation operator along a path to e w m up to time s note hs w is only well defined up to v equivalence for c e w m let xh w denote the vector field along a given by xsh c hs cw h s for each s e 0 1 one should interpret xh as a vector field on w m the vector field xh induces a flow sh t w m w m which leaves wiener measure v quasi invariant see driver 13 it is shown in this paper that the same result is valid if h l 0 and the wiener measure v is replaced by a pinned wiener measure ive the measure ve is proportional to the measure v conditioned on the set of paths which start at 0 e m and end at a fixed end point e e m also as in 13 one gets an integration by parts formula for the vector fields xh defined above

blow up and global existence of solutions to an inhomogeneous parabolic system abstract we study the critical phenomenon of a weakly coupled inhomogeneous parabolic system which is largely unexplored even though the homogeneous problem has been widely studied the conditions on the powers of the nonlinear terms are new compared with the well known homogeneous case the method appears to be different from the standard ones in the literature an application to the lame emden system is also given

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

heat kernel and curvature bounds in ricci flows with bounded scalar curvature abstract in this paper we analyze ricci flows on which the scalar curvature is globally or locally bounded from above by a uniform or time dependent constant on such ricci flows we establish a new time derivative bound for solutions to the heat equation based on this bound we solve several open problems 1 distance distortion estimates 2 the existence of a cutoff function 3 gaussian bounds for heat kernels and 4 a backward pseudolocality theorem which states that a curvature bound at a later time implies a curvature bound at a slightly earlier time using the backward pseudolocality theorem we next establish a uniform l 2 curvature bound in dimension 4 and we show that the flow in dimension 4 converges to an orbifold at a singularity we also obtain a stronger e regularity theorem for ricci flows this result is particularly useful in the study of kahler ricci flows on fano manifolds where it can be used to derive certain convergence results

heat kernel on smooth metric measure spaces with nonnegative curvature we derive a local gaussian upper bound for the f heat kernel on complete smooth metric measure space m g e f dv with nonnegative bakry emery ricci curvature as applications we obtain a sharp l f 1 liouville theorem for f subharmonic functions and an l f 1 uniqueness property for nonnegative solutions of the f heat equation assuming f is of at most quadratic growth in particular any l f 1 integrable f subharmonic function on gradient shrinking and steady ricci solitons must be constant we also provide explicit f heat kernel for gaussian solitons

harnack estimates for a heat type equation under the ricci flow abstract in this paper we consider the gradient estimates for a positive solution of the nonlinear parabolic equation t u t u h u p on a riemannian manifold whose metrics evolve under the ricci flow two harnack inequalities and other interesting results are obtained

guest editorial special issue on positive polynomials in control the 16 papers in this special section focus on positive polynomials in control these papers propose new contributions to the characterization of positive polynomials and illustrate novel applications in the areas of uncertain systems nonlinear systems time delay systems hybrid systems and model predictive control

path integral formulae for heat kernels and their derivatives the heat kernel and its derivatives of a vector laplacian on the sections of a bundle over a compact riemannian manifold are expressed as products of the scalar heat kernel of the manifold and path integrals over the brownian bridge the small time asymptotics of these integrals are computed

on the finiteness property for rational matrices in this chapter we analyze a recent conjecture stating that the finiteness property holds for pairs of binary matrices we show that the finiteness property holds for all pairs of binary matrices if and only if it holds for all sets of nonnegative rationalmatrices we provide a similar result for matrices with positive and negative entries we finally prove the conjecture for 2 2 matrices

analysis of a two phase field model for the solidification of an alloy abstract in this paper we present some theoretical results for a system of nonlinear partial differential equations that provide a phase field model for the solidification melting of a metallic alloy it is assumed that two different kinds of crystallization are possible consequently the unknowns are the temperature and the phase field functions u and v the time derivatives u t and v t appear in the equation for the heat equation on the other hand the equations for u and v contain nonlinear terms where we find

a harnack type inequality for non symmetric markov semigroups abstract for a strong feller and irreducible markov semigroup on a locally compact polish space the harnack type inequality 1 1 holds if and only if the semigroup has a unique invariant probability measure and is ultracontractive moreover new sufficient conditions for this inequality to hold as well as upper bound estimates of the underlying constant are presented for diffusion semigroups on riemannian manifolds

a matrix li yau hamilton estimate for kahler ricci flow in this paper we prove a new matrix li yau hamilton lyh estimate for kahler ricci flow on manifolds with nonnegative bisectional curvature the form of this new lyh estimate is obtained by the interpolation consideration originated in ch by chow this new inequality is shown to be connected with perelman s entropy formula through a family of differential equalities in the rest of the paper we show several applications of this new estimate and its corresponding estimate for linear heat equation these include a sharp heat kernel comparison theorem generalizing the earlier result of li and tian lt a manifold version of stoll s theorem st on the characterization of algebraic divisors and a localized monotonicity formula for analytic subvarieties which sharpens the bishop volume comparison theorem motivated by the connection between the heat kernel estimate and the reduced volume monotonicity of perelman p we prove a sharp lower bound heat kernel estimate for the time dependent heat equation which is in a certain sense dual to perelman s monotonicity of the reduced volume as an application of this new monotonicity formula we show that the blow down limit of a certain type long time solution is a gradient expanding soliton we also illustrate the connection between the new lyh estimate and the hessian comparison theorem of fin on the forward reduced distance local monotonicity formulae on entropy and forward reduced volume are also derived

self interacting diffusions this paper is concerned with a general class of self interacting diffusions x t t 0 living on a compact riemannian manifold m these are solutions to stochastic differential equations of the form dx t brownian increments drift term depending on x t and t the normalized occupation measure of the process it is proved that the asymptotic behavior of t can be precisely related to the asymptotic behavior of a deterministic dynamical semi flow t t 0 defined on the space of the borel probability measures on m in particular the limit sets of t are proved to be almost surely attractor free sets for these results are applied to several examples of self attracting repelling diffusions on the n sphere for instance in the case of self attracting diffusions our results apply to prove that t can either converge toward the normalized riemannian measure or to a gaussian measure depending on the value of a parameter measuring the strength of the attraction

gradient estimates for the heat equation under the ricci flow the paper considers a manifold m evolving under the ricci flow and establishes a series of gradient estimates for positive solutions of the heat equation on m

a new approach to multi phase formulation for the solidification of alloys abstract this paper demonstrates that the standard approach to the modelling of multi phase field dynamics for the solidification of alloys has three major defects and offers an alternative approach the phase field formulation of solidification for alloys with multiple solid phases is formed by relating time derivatives of each variable of the system e g phases and alloy concentration to the variational derivative of free energy with respect to that variable in such a way as to ensure positive local entropy production contributions to the free energy include the free energy density which drives the system and a penalty term for the phase field gradients which ensures continuity in the variables the phase field equations are supplemented by a constraint guaranteeing that at any point in space and time the phases sum to unity how this constraint enters the formulation is the subject of this paper which postulates and justifies an alternative to current methods

sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative ricci curvature in this paper we consider complete noncompact riemannian manifolds m g with nonnegative ricci curvature and euclidean volume growth of dimension n ge 3 for every bounded open subset omega subset m with smooth boundary we prove that begin aligned int limits partial omega left frac mathrm h n 1 right n 1 mathrm d sigma ge mathrm avr g big mathbb s n 1 big end aligned where mathrm h is the mean curvature of partial omega and mathrm avr g is the asymptotic volume ratio of m g moreover the equality holds true if and only if m setminus omega g is isometric to a truncated cone over partial omega an optimal version of huisken s isoperimetric inequality for 3 manifolds is obtained using this result finally exploiting a natural extension of our techniques to the case of parabolic manifolds we also deduce an enhanced version of kasue s non existence result for closed minimal hypersurfaces in manifolds with nonnegative ricci curvature

a converse sum of squares lyapunov result with a degree bound although sum of squares programming has been used extensively over the past decade for the stability analysis of nonlinear systems several fundamental questions remain unanswered in this paper we show that exponential stability of a polynomial vector field on a bounded set implies the existence of a lyapunov function which is a sum of squares of polynomials in particular the main result states that if a system is exponentially stable on a bounded nonempty set then there exists a sum of squares lyapunov function which is exponentially decreasing on that bounded set furthermore we derive a bound on the degree of this converse lyapunov function as a function of the continuity and stability properties of the vector field the proof is constructive and uses the picard iteration our result implies that semidefinite programming can be used to answer the question of stability of a polynomial vector field with a bound on complexity

recurrence and transience of branching diffusion processes on riemannian manifolds we relate the recurrence and transience of a branching diffusion process on a riemannian manifold m to some properties of a linear elliptic operator onm including spectral properties there is a trade off between the tendency of the transient brownian motion to escape and the birth process of the new particles if the latter has a high enough intensity then it may override the transience of the brownian motion leading to the recurrence of the branching process and vice versa in the case of a spherically symmetric manifold the critical intensity of the population growth can be found explicitly

the global behavior of heat kernels in exterior domains abstract we establish the global boundary behavior of dirichlet heat kernels on some unbounded domains these combined with the interior estimate in the recent paper by grigoryan and saloff coste comm pure appl math 55 1 2002 93 provide a complete and qualitatively sharp description of heat kernels g of dirichlet laplacians on some unbounded c1 1 domains d m here m is certain complete noncompact manifold with nonnegative ricci curvature for example there exist positive constants c1 c2 depending on the unbounded domain d m such that for x dist x d x t 1 1 y t 1 1 c 1 e c 2 d x y 2 t b x t g x t y 0 x t 1 1 y t 1 1 e d x y 2 c 2 t c 1 b x t for all x y d and t 0 existence of these global bounds had been an unresolved problem even in the euclidean case

a globally asymptotically stable polynomial vector field with rational coefficients and no local polynomial lyapunov function abstract we give an explicit example of a two dimensional polynomial vector field of degree seven that has rational coefficients is globally asymptotically stable but does not admit an analytic lyapunov function even locally

lifschitz tail on hyperbolic space neumann conditions on considere un espace hyperbolique h de dimension n et l operateur laplacien hyperbolique h etant donne une distribution aleatoire de points a l aide d un processus ponctuel de poisson tel que chaque point se situe dans une boule hyperbolique de rayons on etudie la question suivante si est un petit nombre reel combien y a t il de valeurs propres inferieures a pour l operateur 1 2 h dans une boule b de centre o avec des conditions de neumann sur la frontiere et avec des conditions aux limites de dirichlet sur les boules aleatoires formant des intersections avec b si le rayon de b est suffisamment grand

the ricci flow on the 2 sphere the classical uniformization theorem interpreted differential geomet rically states that any riemannian metric on a 2 dimensional surface ispointwise conformal to a constant curvature metric thus one can con sider the question of whether there is a natural evolution equation whichconformally deforms any metric on a surface to a constant curvature met ric the primary interest in this question is not so much to give a newproof of the uniformization theorem but rather to understand nonlinearparabolic equations better especially those arising in differential geome try a sufficiently deep understanding of parabolic equations should yieldimportant new results in riemannian geometry the question in the preceding paragraph has been studied by richardhamilton 3 and brad osgood ralph phillips and peter sarnak 6 in 3 hamilton studied the following equation which we refer to as

extremal of log sobolev inequality and w entropy on noncompact manifolds abstract let m be a complete connected noncompact manifold with bounded geometry under a condition near infinity we prove that the log sobolev functional 1 1 has an extremal function decaying exponentially near infinity we also prove that an extremal function may not exist if the condition is violated this result has the following consequences 1 it seems to give the first example of connected complete manifolds with bounded geometry where a standard log sobolev inequality does not have an extremal 2 it gives a negative answer to the open question on the existence of extremal of perelman s w entropy in the noncompact case which was stipulated by perelman 2002 22 p 9 3 2 remark 3 it helps to prove in some cases that noncompact shrinking breathers of ricci flow are gradient shrinking solitons

surfaces with boundary their uniformizations determinants of laplacians and isospectrality let be a compact surface of type g n n 0 obtained by re moving n disjoint disks from a closed surface of genus g assuming 0 while osgood phillips and sarnak o p s3 showed the properness when g 0

global boundedness in a quasilinear attraction repulsion chemotaxis model with nonlinear sensitivity abstract this paper considers the attraction repulsion chemotaxis model with homogeneous neumann boundary conditions in a smooth bounded convex domain in this model when the scaling constant is zero and the chemotactic sensitivity functions are nonlinear we prove that this system possesses a unique global classical solution that is uniformly bounded under some assumptions when the scaling constant is one and one of the chemotactic sensitivity functions is nonlinear we also obtain a unique bounded global classical solution under some assumptions

the finiteness conjecture for the generalized spectral radius of a set of matrices the generalized spectral radius g9 of a set of n n matrices is g9 lim supk g9k 1k where g9k sup a1a2 ak each ai the joint spectral radius g9 is g9 lim supk g9k 1k where g9k sup a1 ak each ai it is known that g9 g9 holds for any finite set of n n matrices the finiteness conjecture asserts that for any finite set of real n n matrices there exists a finite k such that g9 g9 g9k 1k the normed finiteness conjecture for a given operator norm asserts that for any finite set a1 am having all ai op 1 either g9 1 or g9 g9 g9k 1k 1 for some finite k it is shown that the finiteness conjecture is true if and only if the normed finiteness conjecture is true for all operator norms the normed finiteness conjecture is proved for a large class of operator norms extending results of gurvits in particular for polytope norms and for the euclidean norm explicit upper bounds are given for the least k having g9 g9k 1k these results imply upper bounds for generalized critical exponents for these norms

harnack estimates for nonlinear backward heat equations in geometric flows let m be a closed riemannian manifold with a family of riemannian metrics gij t evolving by a geometric flow tgij 2sij where sij t is a family of smooth symmetric two tensors we derive several differential harnack estimates for positive solutions to the nonlinear backward heat type equation f t f flog f asf where a and are constants and s gijsij is the trace of sij our abstract formulation provides a unified framework for some known results proved by various authors and moreover leads to new harnack inequalities for a variety of geometric flows

compact manifolds with a little negative curvature on etend le theoreme de bochner et le theoreme de myers pour des p formes en donnant des obstructions topologiques pour des varietes admettant des metriques avec une petite courbure negative

lmi techniques for optimization over polynomials in control a survey numerous tasks in control systems involve optimization problems over polynomials and unfortunately these problems are in general nonconvex in order to cope with this difficulty linear matrix inequality lmi techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems this survey aims to provide the reader with a significant overview of the lmi techniques that are used in control systems for tackling optimization problems over polynomials describing approaches such as decomposition in sum of squares positivstellensatz theory of moments polya s theorem and matrix dilation moreover it aims to provide a collection of the essential problems in control systems where these lmi techniques are used such as stability and performance investigations in nonlinear systems uncertain systems time delay systems and genetic regulatory networks it is expected that this survey may be a concise useful reference for all readers

on anisotropic order parameter models for multi phase system and their sharp interface limits abstract for a general class of diffuse anisotropic multi phase order parameter or phase field models we use formally matched asymptotic expansions to determine the asymptotic limit when a small parameter related to the thickness of the interface tends to zero in the case of anisotropic allen cahn systems we obtain in the limit that the interface moves by anisotropic mean curvature flow at triple junctions a force balance holds which in the anisotropic case includes shear forces herring torque terms acting normal to the interface we further identify the singular limit of anisotropic cahn hilliard systems

new li yau hamilton inequalities for the ricci flow via the space time approach we prove li yau hamilton inequalties that extend hamilton s matrix inequality for solutions of the ricci flow with nonnegative curvature operators to obtain our extensions we apply the space time formalism of s c chu and one of the authors to solutions of the ricci flow modified by a cosmological constant then we adjoin to the ricci flow the evolution of a 1 form and a 2 form flowing by a system of heat type equations by a rescaling argument the inequalities we obtain in this manner yield new inequalities which are reminiscent of the linear trace inequality of hamilton and one of the authors

anisotropic multi phase field model interfaces and junctions n this paper we bring together and extend two recent developments in phase field models namely a phase field model of a multiphase system i steinbach et al physica d 94 135 1996 and the extension of the cahn hoffman vector theory of anisotropic sharp interfaces to phase field models a a wheeler and g b mcfadden eur j appl math 7 369 1996 proc r soc london ser a 453 1611 1997 we develop the phase field model of a multiphase system proposed by steinbach et al to include both surface energy and interfacial kinetic anisotropy we show that this model may be compactly expressed in terms of generalized cahn hoffman vectors this generalized cahn hoffman vector formalism is subsequently developed to include the notion of a stress tensor which is used to succinctly derive the leading order conditions at both moving interfaces and stationary multijunctions in the sharp interface limit

dimension dependent hypercontractivity for gaussian kernels we derive sharp local and dimension dependent hypercontractive bounds on the markov kernel of a large class of diffusion semigroups unlike the dimension free ones they capture refined properties of markov kernels such as trace estimates they imply classical bounds on the ornstein uhlenbeck semigroup and a dimensional and refined transportation talagrand inequality when applied to the hamilton jacobi equation hypercontractive bounds on the ornstein uhlenbeck semigroup driven by a non diffusive levy semigroup are also investigated curvature dimension criteria are the main tool in the analysis

sharp differential estimates of li yau hamilton type for positive p p forms on kahler manifolds in this paper we study the heat equation of hodge laplacian deformation of p p forms on a kahler manifold after identifying the condition and establishing that the positivity of a p p form solution is preserved under such an invariant condition we prove the sharp differential harnack in the sense of liyau hamilton estimates for the positive solutions of the hodge laplacian heat equation we also prove a nonlinear version coupled with the kahler ricci flow and some interpolating matrix differential harnack type estimates for both the kahler ricci flow and the ricci flow 2011 wiley periodicals inc

integrated density of states for random metrics on manifolds we study ergodic random schrodinger operators on a covering manifold where the randomness enters both via the potential and the metric we prove measurability of the random operators almost sure constancy of their spectral properties the existence of a self averaging integrated density of states and a pastur ubin type trace formula

remarks on li yau inequality on graphs abstract in this paper we study li yau gradient estimates for the solutions u to the heat equation t u u on graphs under the curvature condition c d n k which is introduced by bauer et al in 5 as applications we derive harnack inequalities and heat kernel estimates on graphs also we present a type of hamilton gradient estimates

a generalized field method for multiphase transformations using interface fields abstract the recently developed multiphase field method describing the interaction between an arbitrary number of individual phase fields with individual characteristics is reformulated by the use of interface fields this reformulation allows for the decomposition of the nonlinear multiphase field interactions into pairwise interaction of interface fields this removes some difficulties in the treatment of triple points or higher order interactions that occurred in the original model the interface fields being defined in a 2 n dimensional space where n is the order of the multiple point can be interpreted being the generalized coordinates for this variational problem the considered example of a multiphase change problem indicates clearly that a relaxation ansatz for the evolution of the field variables towards the minimum of the free energy is warranted only for generalized coordinates while a relaxation ansatz using functionally dependent variables and the lagrange formalism in general mixes time and energy scales

exponentially stable nonlinear systems have polynomial lyapunov functions on bounded regions this paper presents a proof that existence of a polynomial lyapunov function is necessary and sufficient for exponential stability of a sufficiently smooth nonlinear vector field on a bounded set the main result states that if there exists an n times continuously differentiable lyapunov function which proves exponential stability on a bounded subset of rn then there exists a polynomial lyapunov function which proves exponential stability on the same region such a continuous lyapunov function will exist if for example the vector field is at least n times continuously differentiable the proof is based on a generalization of the weierstrass approximation theorem to differentiable functions in several variables specifically polynomials can be used to approximate a differentiable function using the sobolev norm w 1 infin to any desired accuracy this approximation result is combined with the second order taylor series expansion to show that polynomial lyapunov functions can approximate continuous lyapunov functions arbitrarily well on bounded sets the investigation is motivated by the use of polynomial optimization algorithms to construct polynomial lyapunov functions

an analytical study of the static state of multi junctions in a multi phase field model abstract we investigate the properties of the multi order parameter phase field model of steinbach and pezzolla i steinbach f pezzolla a generalized field method for multi phase transformations using interface fields physica d 134 1999 385 393 with respect to the behavior in triple and higher order junctions from the structure of this model it was speculated that dynamical solutions may exist in the triple junction which could lead to a violation of young s law here we confirm analytically recent numerical simulations showing that such dynamical states do not exist and that an equilibrium solution therefore does indeed correspond to a minimum of the free energy this implies that young s law must be satisfied in the framework of the model we show that young s law is a consequence of the interface kinetic equilibrium and not due to a mechanical force balance in agreement with earlier predictions c caroli c misbah on static and dynamical young s condition at a trijunction j phys i france 7 1997 1259 1265

an explicit counterexample to the lagarias wang finiteness conjecture abstract the joint spectral radius of a finite set of real d d matrices is defined to be the maximum possible exponential rate of growth of long products of matrices drawn from that set a set of matrices is said to have the finiteness property if there exists a periodic product which achieves this maximal rate of growth j c lagarias and y wang conjectured in 1995 that every finite set of real d d matrices satisfies the finiteness property however t bousch and j mairesse proved in 2002 that counterexamples to the finiteness conjecture exist showing in particular that there exists a family of pairs of 2 2 matrices which contains a counterexample similar results were subsequently given by v d blondel j theys and a a vladimirov and by v s kozyakin but no explicit counterexample to the finiteness conjecture has so far been given the purpose of this paper is to resolve this issue by giving the first completely explicit description of a counterexample to the lagarias wang finiteness conjecture namely for the set a 1 1 0 1 1 0 1 1 we give an explicit value of 0 749326546330367557943961948091344672091327370236064317358024 such that a does not satisfy the finiteness property