hamiltonian stationary cones and self similar solutions in higher dimension in lw we construct examples of two dimensional hamiltonian stationary self shrinkers and self expanders for lagrangian mean curvature flows which are asymptotic to the union of two schoen wolfson cones these self shrinkers and self expanders can be glued together to yield solutions of the brakke flow a weak formulation of the mean curvature flow moreover there is no mass loss along the brakke flow in this paper we generalize these results to higher dimension we construct new higher dimensional hamiltonian stationary cones of different topology as generalizations of the schoen wolfson cones hamiltonian stationary self shrinkers and self expanders that are asymptotic to these hamiltonian stationary cones are also constructed they can also be glued together to produce eternal solutions of the brakke flow without mass loss finally we show the same conclusion holds for those lagrangian self similar examples recently found by joyce tsui and the first author in jlt

adaptive artificial boundary condition for the two level schrodinger equation with conical crossings in this paper we present an adaptive approach to design the artificial boundary conditions for the two level schrodinger equation with conical crossings on the unbounded domain we use the windowed fourier transform to obtain the local wave number information in the vicinity of artificial boundaries and adopt the operator splitting method to obtain an adaptive local artificial boundary condition then reduce the original problem into an initial boundary value problem on the bounded computational domain which can be solved by the finite difference method by this numerical method we observe the surface hopping phenomena of the two level schrodinger equation with conical crossings several numerical examples are provided to show the accuracy and convergence of the proposed method

artificial boundary conditions for one dimensional cubic nonlinear schrodinger equations this paper addresses the construction of nonlinear integro differential artificial boundary conditions for one dimensional nonlinear cubic schrodinger equations several ways of designing such conditions are provided and a theoretical classification of their accuracy is given semidiscrete time schemes based on the method developed by duran and sanz serna ima j numer anal 20 2000 pp 235 261 are derived for these unusual boundary conditions stability results are stated and several numerical tests are performed to analyze the capacity of the proposed approach

absorbing boundary conditions for one dimensional nonlinear schrodinger equations we construct two families of absorbing boundary conditions for the nonlinear schrodinger equation the first one relies on the pseudodifferential calculus and the second one relies on the paradifferential calculus we show that some of the corresponding initial boundary value problems are well posed we finally present numerical experiments illustrating the efficiency of these methods

absorbing boundary conditions for the schrodinger equation a large number of differential equation problems which admit traveling waves are usually defined on very large or infinite domains to numerically solve these problems on smaller subdomains of the original domain artificial boundary conditions must be defined for these subdomains one type of artificial boundary condition which can minimize the size of such subdomains is the absorbing boundary condition the imposition of absorbing boundary conditions is a technique used to reduce the necessary spatial domain when numerically solving partial differential equations that admit traveling waves such absorbing boundary conditions have been extensively studied in the context of hyperbolic wave equations in this paper general absorbing boundary conditions will be developed for the schrodinger equation with one spatial dimension using group velocity considerations previously published absorbing boundary conditions will be shown to reduce to special cases of this absorbing boundary condition the well posedness of the initial boundary value problem of the absorbing boundary condition coupled to the interior schrodinger equation will also be discussed extension of the general absorbing boundary condition to higher spatial dimensions will be demonstrated numerical simulations using initial single gaussian double gaussian and a narrow gaussian pulse distributions will be given with comparision to exact solutions to demonstrate the reflectivity properties of various orders of the absorbing boundary condition

absorbing boundary conditions for difference approximations to the multi dimensional wave equation construction des conditions aux limites absorbantes pour l equation des ondes multidimensionnelle a l aide d approximations aux differences formulation sous forme canonique des conditions aux limites absorbantes analytiques

on the box counting dimension of graphs of harmonic functions on the sierpinski gasket abstract the objective of this paper is to study the box counting dimension of graphs of fractal interpolation functions and harmonic functions on the sierpinski gasket firstly we give construction of a fractal interpolation function on the sierpinski gasket and then with the help of fractal interpolation functions we show the existence of fractal functions in the space dom e consisting of all finite energy functionals on the sierpinski gasket later we provide bounds for the box counting dimension of graphs of some functions belonging to the family of continuous functions which arise as fractal interpolation functions moreover we also obtain bounds for the box counting dimension of graphs of harmonic functions and piecewise harmonic functions also we obtain upper and lower bounds for the box counting dimension of graphs of functions that belong to dom e

some properties of laplacians on fractals abstract kigami has defined an analog of the laplacian on a class of self similar fractals including the familiar sierpinski gasket we study properties of this operator we show that there is a maximal principle for solutions of certain nonlinear equations of the form u x f x u x we discuss the extension of the laplacian to non compact fractal blow ups and show that it is essentially self adjoint and we prove an analog of liouville s theorem in some cases we also give an explicit algorithm for solving the dirichlet problem for certain domains in the sierpinski gasket and give a characterization of all harmonic functions on those domains

what is not in the domain of the laplacian on sierpinski gasket type fractals abstract we consider the analog of the laplacian on the sierpinski gasket and related fractals constructed by kigami a function f is said to belong to the domain of if f is continuous and f is defined as a continuous function we show that if f is a nonconstant function in the domain of then f2 is not in the domain of we give two proofs of this fact the first is based on the analog of the pointwise identity f2 2f f f 2 where we show that f 2 does not exist as a continuous function in fact the correct interpretation of f2 is as a singular measure a result due to kusuoka we give a new proof of this fact the second is based on a dichotomy for the local behavior of a function in the domain of at a junction point x0 of the fractal in the typical case nonvanishing of the normal derivative we have upper and lower bounds for f x f x0 in terms of d x x0 for a certain value and in the nontypical case vanishing normal derivative we have an upper bound with an exponent greater than 2 this method allows us to show that general nonlinear functions do not operate on the domain of

wigner measures and codimension two crossings this article gives a semiclassical description of nucleonic propagation through codimension two crossings of electronic energy levels codimension two crossings are the simplest energy level crossings which affect the born oppenheimer approximation in the zeroth order term the model we study is a two level schrodinger equation with a laplacian as kinetic operator and a matrix valued linear potential whose eigenvalues cross if the two nucleonic coordinates equal zero we discuss the case of well localized initial data and obtain a description of the wavefunction s two scaled wigner measure and of the weak limit of its position density which is valid globally in time

on the existence of three solutions for the dirichlet problem on the sierpinski gasket we apply a recently obtained three critical point theorem of b ricceri to prove the existence of at least three solutions of certain two parameter dirichlet problems defined on the sierpinski gasket we also show the existence of at least three nonzero solutions of certain perturbed two parameter dirichlet problems on the sierpinski gasket using both the mountain pass theorem of ambrosetti and rabinowitz and that of pucci and serrin

generalized hamiltonian structures for ermakov systems we construct poisson structures for ermakov systems using the ermakov invariant as the hamiltonian two classes of poisson structures are obtained one of them is degenerate in which case we derive the casimir functions in some situations the existence of casimir functions can give rise to superintegrable ermakov systems finally we characterize the cases where linearization of the equations of motion is possible

singularly continuous spectrum of a self similar laplacian on the half line we investigate the spectrum of the self similar laplacian which generates the so called pq random walk on the integer half line using the method of spectral decimation we prove that the spectral type of the laplacian is singularly continuous whenever p 12 this serves as a toy model for generating singularly continuous spectrum which can be generalized to more complicated settings we hope it will provide more insight into fibonacci type and other weakly self similar models

lagrangian homology classes without regular minimizers we show that there is an integral homology class in a kahler einstein surface that can be represented by a lagrangian two sphere but that a minimizer of area among lagrangian two spheres representing this class has isolated singularities with non flat tangent cones

semilinear equations on fractal blowups abstract the paper concerns existence of a ground state for a nonlinear scalar field equation on a blowup fractal where imbedding of the energy space into l p is not compact in absence of invariant transformations involved in conventional concentration compactness argument the paper develops convergence reasoning based on the fractal s self similarity

hamiltonian stationary shrinkers and expanders for lagrangian mean curvature flows we construct examples of shrinkers and expanders for lagrangian mean curvature flows these examples are hamiltonian stationary and asymptotic to the union of two hamiltonian stationary cones found by schoen and wolfson in sw the schoen wolfson cones cp q are obstructions to the existence problems of special lagrangians or lagrangian minimal surfaces in the variational approach it is known that these cone singularities cannot be resolved by any smooth oriented lagrangian submanifolds the shrinkers and expanders that we found can be glued together to yield solutions of the brakke motiona weak formulation of the mean curvature flow for any coprime pair p q other than 2 1 we construct such a solution that resolves any single schoen wolfson cone cp q this thus provides an evidence to schoen wolfson s conjecture that the 2 1 cone is the only area minimizing cone higher dimensional generalizations are also obtained

brownian motion on the sierpinski gasket we construct a brownian motion taking values in the sierpinski gasket a fractal subset of 2 and study its properties this is a diffusion process characterized by local isotropy and homogeneity properties we show for example that the process has a continuous symmetric transition density p t x y with respect to an appropriate hausdorff measure and obtain estimates on p t x y

on the lowest eigenvalue of the laplacian for the intersection of two domains ifa andb are two bounded domains in n and a b are the lowest eigenvalues of with dirichlet boundary conditions then there is some translate b x ofb such that a b x a b a similar inequality holds for lambda p a inf parallel nabla f parallel p p parallel f parallel p p f in w 0 1 p a there are two corollaries of this theorem i a lower bound for sup x volume a b x in terms of a whenb is a ball ii a compactness lemma for certain sequences inw 1 p n

brownian motion penetrating fractals an application of the trace theorem of besov spaces abstract for a closed connected set f in rn assume that there is a local regular dirichlet form a symmetric diffusion process on f whose domain is included in a lipschitz space or a besov space on f under some condition for the order of the space and the newtonian 1 capacity of f we prove that there exists a symmetric diffusion process on rn which moves like the process on f and like brownian motion on rn outside f as an application we will show that when f is a nested fractal or a sierpinski carpet whose hausdorff dimension is greater than n 2 we can construct brownian motion penetrating the fractal for the proof we apply the technique developed in the theory of besov spaces

harmonization and homogenization on fractals this paper suggests a direct approach to define the laplacian the spectral dimension of nested fractals and the pre sierpinski carpet conductivity we find a geometric construction of the harmonic functions on the gasket and therefore can describe effectively the dense set of functions having finite energy the paper is mostly aimed at the homogenization on the pre sierpinski gasket whose horizontal and nonhorizontal bonds have different conductivities a andb respectively we prove the convergence of the rescaled energies on the pre sierpinski gasket to a b e where e is the standard energy on the gasket with uniform conductivities we also find an explicit expression for the effective conductivity a b and deduce that its set of singularities turns out to be the julia set of a certain rational function a special section is devoted to the problem of the pre sierpinski carpet conductivity asymptotic behavior for this problem a new proof of barlow bass inequalities with sharper constants is given

pure point spectrum for the laplacian on unbounded nested fractals abstract we prove for the class of nested fractals introduced by t lindstrom 1990 memoirs amer math soc 420 that the integrated density of states is completely created by the so called neuman dirichlet eigenvalues the corresponding eigenfunctions lead to eigenfunctions with compact support on the unbounded set and we prove that for a large class of blow ups the set of neuman dirichlet eigenfunctions is complete leading to a pure point spectrum with compactly supported eigenfunctions this generalizes previous results of h teplyaev 1998 j funct anal 159 537 567 on the sierpinski gasket our methods are elementary and use only symmetry arguments via the representations of the symmetry group of the set

weak uncertainty principle for fractals graphs and metric measure spaces we develop a new approach to formulate and prove the weak uncertainty inequality which was recently introduced by okoudjou and strichartz we assume either an appropriate measure growth condition with respect to the effective resistance metric or in the absence of such a metric we assume the poincare inequality and reverse volume doubling property we also consider the weak uncertainty inequality in the context of nash type inequalities our results can be applied to a wide variety of metric measure spaces including graphs fractals and manifolds

brownian motion on a random recursive sierpinski gasket we introduce a random recursive fractal based on the sierpinski gasket and construct a diffusion upon the fractal via a dirichlet form this form and its symmetrizing measure are determined by the electrical resistance of the fractal the effective resistance provides a metric with which to discuss the properties of the fractal and the diffusion the main result is to obtain uniform upper and lower bounds for the transition density of the brownian motion on the fractal in terms of this metric the bounds are not tight as there are logarithmic corrections due to the randomness in the environment and the behavior of the shortest paths in the effective resistance metric is not well understood the results are deduced from the study of a suitable general branching process

semilinear pdes on self similar fractals a laplacian may be defined on self similar fractal domains in terms of a suitable self similar dirichlet form enabling discussion of elliptic pdes on such domains in this context it is shown that that semilinear equations such as u u p 0 with zero dirichlet boundary conditions have non trivial non negative solutions if 0 1 or if 2 and 1 p 2 2 where is the intrinsic dimension or spectral dimension of the system thus the intrinsic dimension takes the r o le of the euclidean dimension in the classical case in determining critical exponents of semilinear problems

harmonic calculus on fractals a measure geometric approach ii differentiation of functions w r t finite atomless measures with compact support on the real line is introduced the related harmonic calculus is similar to that of the classical lebesgue case as an application we obtain the weyl exponent for the spectral asymptotics of the laplacians w r t linear cantor type measures with arbitrary weights

non linear elliptical equations on the sierpinski gasket this paper investigates properties of certain nonlinear pdes on fractal sets with an appropriately defined laplacian we obtain a number of results on the existence of non trivial solutions of the semilinear elliptic equation u a x u f x u with zero dirichlet boundary conditions where u is defined on the sierpinski gasket we use the mountain pass theorem and the saddle point theorem to study such equations for different classes of a and f a strong sobolev type inequality leads to properties that contrast with those for classical domains

on short time existence of lagrangian mean curvature flow we consider a short time existence problem motivated by a conjecture of joyce conjectures on bridgeland stability for fukaya categories of calabi yau manifolds special lagrangians and lagrangian mean curvature flow arxiv 1401 4949 2014 specifically we prove that given any compact lagrangian l subset mathbb c n with a finite number of singularities each asymptotic to a pair of non area minimising transversally intersecting lagrangian planes there is a smooth lagrangian mean curvature flow existing for some positive time that attains l as t searrow 0 as varifolds and smoothly locally away from the singularities

analysis on products of fractals for a class of post critically finite p c f fractals which includes the sierpinski gasket sg there is a satisfactory theory of analysis due to kigami including energy harmonic functions and laplacians in particular the laplacian coincides with the generator of a stochastic process constructed independently by probabilistic methods the probabilistic method is also available for non p c f fractals such as the sierpinski carpet in this paper we show how to extend kigami s construction to products of p c f fractals since the products are not themselves p c f this gives the first glimpse of what the analytic theory could accomplish in the non p c f setting there are some important differences that arise in this setting it is no longer true that points have positive capacity so functions of finite energy are not necessarily continuous also the boundary of the fractal is no longer finite so boundary conditions need to be dealt with in a more involved manner all in all the theory resembles pde theory while in the p c f case it is much closer to ode theory

transition density estimates for diffusion processes on post critically finite self similar fractals the recent development of analysis on fractal spaces is physically motivated by the study of diffusion in disordered media the natural questions that arise concern the existence and uniqueness of a suitable laplace operator and the behaviour of the associated heat semigroup on a space which is fractal the classes of fractals for which these questions were rst answered were classes of exactly self similar fractals with strong spatial symmetry such as nested or af ne nested fractals see for example 2 8 the existence of a laplacian and estimates on the heat kernel were obtained by considering the associated diffusion process and using the symmetry of the space the uniqueness of the laplacian for nested and af ne nested fractals has recently been solved through consideration of their dirichlet forms 23 in 15 the framework of post critically nite which we abbreviate to p c f self similar sets was introduced in order to capture the notion of exactly selfsimilar nitely rami ed fractals as used in the physics literature finitely rami ed fractals have the property that the intersection of any connected subset of the fractal with the rest of the set should occur only at a nite number of points this makes these structures much easier to analyse than in nitely rami ed sets such as the sierpinski carpet 5 the p c f self similar sets do not have spatial symmetry in general and have provided a mathematical test bed for analysis on fractals in this paper we will obtain uniform short time estimates on the heat kernel associated with a natural laplacian on the fractal a laplacian can be constructed on a p c f self similar fractal as a limit of discrete laplacians on graph approximations to the fractal based on the rami cation points we construct these operators via their dirichlet forms which can be set on any l space with a full measure n in 23 criteria are given which give a partial answer to the existence and uniqueness of laplace type operators on p c f self similar sets in a series of papers 16 17 18 20 25 7 some of the interesting spectral properties of p c f sets have been elucidated it has been shown that there can exist localized eigenfunctions if there is a high degree of symmetry in the set and corresponding laplace operator this corresponds to oscillation in the leading order term for the asymptotics of the eigenvalue counting function such behaviour occurs for nested fractals 7 and p c f sets with strong harmonic

monte carlo sampling of wigner functions and surface hopping quantum dynamics the article addresses the achievable accuracy for a monte carlo sampling of wigner functions in combination with a surface hopping algorithm for non adiabatic quantum dynamics the approximation of wigner functions is realized by an adaption of the metropolis algorithm for real valued functions with disconnected support the integration which is necessary for computing values of the wigner function uses importance sampling with a gaussian weight function the numerical experiments agree with theoretical considerations and show an error of 2 3

nonlinear wave equations on a class of bounded fractal sets the nonlinear wave equation u tt u f u with given initial data and zero boundary conditions on a class of bounded self similar fractal sets is investigated the sobolev type inequality is the starting point of this work which holds for a class of fractals including the well known sierpinski gasket we obtain the global existence of strong solutions for suitable f if the spectral dimension ds of the fractal satisfies ds 2 the key is to construct the wave propagator and hilbert spaces of functions on the fractal the main difficulty in obtaining the global existence of a weak solution is establishing a priori estimates depending on a regularity property for f t he regularity property of a weak solution is obtained through a fine analysis in which the sobolev type inequality plays a crucial role 2002 elsevier science usa all rights reserved

absorbing boundary conditions for diffusion equations summary we construct and analyse a family of absorbing boundary conditions for diffusion equations with variable coefficients curved artifical boundary and arbitrary convection it relies on the geometric identification of the dirichlet to neumann map and rational interpolation of z 1 2 in the complex plane the boundary conditions are stable accurate and practical for computations

metric based upscaling we consider divergence form elliptic operators in dimension n ge 2 with l coefficients although solutions of these operators are only holder continuous we show that they are differentiable c1 with respect to harmonic coordinates it follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales this new numerical homogenization method is based on the transfer of a new metric in addition to traditional averaged homogenized quantities from subgrid scales into computational scales error bounds can be given and this method can also be used as a compression tool for differential operators 2006 wiley periodicals inc

harmonic calculus on p c f self similar sets the main object of this paper is the laplace operator on a class of fractals first we establish the concept of the renormalization of difference operators on post critically finite p c f for short self similar sets which are large enough to include finitely ramified self similar sets and extend the results for sierpinski gasket given in 10 to this class under each invariant operator for renormalization the laplace operator green function dirichlet form and neumann derivatives are explicitly constructed as the natural limits of those on finite pre self similar sets which approximate the p c f self similar sets also harmonic functions are shown to be finite dimensional and they are characterized by the solution of an infinite system of finite difference equations

plancherel type estimates and sharp spectral multipliers abstract we study general spectral multiplier theorems for self adjoint positive definite operators on l2 x where x is any open subset of a space of homogeneous type we show that the sharp hormander type spectral multiplier theorems follow from the appropriate estimates of the l2 norm of the kernel of spectral multipliers and the gaussian bounds for the corresponding heat kernel the sharp hormander type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the riesz means summability which we also study here we discuss several examples which include sharp spectral multiplier theorems for a class of scattering operators on r3 and new spectral multiplier theorems for the laguerre and hermite expansions

semilinear fractal problems approximation and regularity results abstract we prove existence and uniqueness of the weak solution for a second order semilinear transmission problem in a bounded domain in r 3 in the case of a koch type fractal layer s or the corresponding prefractal layer approximating s we study the asymptotic behavior of the solutions of the corresponding prefractal problem and we prove regularity results for the weak solution via a generalized green formula for domains with fractal boundaries

one parameter dirichlet problems on the sierpinski gasket abstract using a three critical point theorem due to arcoya and carmona we show that under suitable conditions and for sufficiently large real parameters the following nonlinear elliptic equation defined on the sierpinski gasket and with zero dirichlet boundary conditions u x b x u x f x u x has at least two nonzero weak solutions

inhomogeneous parabolic equations on unbounded metric measure spaces kenneth j falconermathematical institute university of st andrews north haugh st andrews fife ky16 9ss uk kjf st andrews ac uk jiaxin hu and yuhua sundepartment of mathematical sciences tsinghua university beijing 100084 people s republic of china hujiaxin mail tsinghua edu cn sunyh08 mails tsinghua edu cn

spectral analysis on infinite sierpinski gaskets we study the spectral properties of the laplacian on infinite sierpin ski gaskets we prove that the laplacian with the neumann boundary condition has pure point spectrum moreover the set of eigenfunctions with compact support is complete the same is true if the infinite sierpin ski gasket has no boundary but is false for the laplacian with the dirichlet boundary condition in all these cases we describe the spectrum of the laplacian and all the eigenfunctions with compact support to obtain these results first we prove certain new formulas for eigenprojectors of the laplacian on finite sierpin ski pre gaskets then we prove that the spectrum of the discrete laplacian on a sierpin ski lattice is pure point and the eigenfunctions are localized 1998 academic press

unconditionally stable discretization schemes of non reflecting boundary conditions for the one dimensional schrodinger equation this paper addresses the problem of the construction of stable approximation schemes for the one dimensional linear schrodinger equation set in an unbounded domain after a study of the initial boundary value problem in a bounded domain with a transparent boundary condition some unconditionally stable discretization schemes are developed for this kind of problem the main difficulty is linked to the involvement of a fractional integral operator defining the transparent operator the proposed semi discretization of this operator yields with a very different point of view the one proposed by yevick friese and schmidt j comput phys 168 2001 433 two possible choices of transparent boundary conditions based on the dirichlet neumann dn and neumann dirichlet nd operators are presented to preserve the stability of the fully discrete scheme conform galerkin finite element methods are employed for the spatial discretization finally some numerical tests are performed to study the respective accuracy of the different schemes

function spaces on fractals abstract we construct function spaces analogs of holder zygmund besov and sobolev spaces on a class of post critically finite self similar fractals in general and the sierpinski gasket in particular based on the laplacian and effective resistance metric of kigami this theory is unrelated to the usual embeddings of these fractals in euclidean space and so our spaces are distinct from the function spaces of jonsson and wallin although there are some coincidences for small orders of smoothness we show that the laplacian acts as one would expect an elliptic pseudodifferential operator of order d 1 on a space of dimension d to act where d is determined by the growth rate of the measure of metric balls we establish some sobolev embedding theorems and some results on complex interpolation on these spaces

propagation through conical crossings an asymptotic semigroup we consider the standard model problem for a conical intersection of electronic surfaces in molecular dynamics our main result is the construction of a semi group in order to approximate the wigner function associated with the solution of the schrodinger equation at leading order in the semiclassical parameter the semigroup stems from an underlying markov process that combines deterministic transport along classical trajectories within the electronic surfaces and random jumps between the surfaces near the crossing our semigroup can be viewed as a rigorous mathematical counterpart of so called trajectory surface hopping algorithms which are of major importance in molecular simulations in chemical physics the key point of our analysis the incorporation of the nonadiabatic transitions is based on the landau zener type formula of fermanian kammerer and gerard10 for the propagation of two scale wigner measures through conical crossings 2005 wiley periodicals inc

adaptive absorbing boundary conditions for schrodinger type equations application to nonlinear and multi dimensional problems we propose an adaptive approach in picking the wave number parameter of absorbing boundary conditions for schrodinger type equations based on the gabor transform which captures local frequency information in the vicinity of artificial boundaries the parameter is determined by an energy weighted method and yields a quasi optimal absorbing boundary conditions it is shown that this approach can minimize reflected waves even when the wave function is composed of waves with different group velocities we also extend the split local absorbing boundary slab method z xu h han phys rev e 74 2006 037704 to problems in multi dimensional nonlinear cases by coupling the adaptive approach numerical examples of nonlinear schrodinger equations in one and two dimensions are presented to demonstrate the properties of the discussed absorbing boundary conditions

finite time singularities for lagrangian mean curvature flow given any embedded lagrangian on a four dimensional compact calabi yau we find another lagrangian in the same hamiltonian isotopy class that develops a finite time singularity under mean curvature flow this contradicts a weaker version of the thomas yau conjecture regarding long time existence and convergence of lagrangian mean curvature flow

dirichlet forms on fractals poincare constant and resistance we study dirichlet forms associated with random walks on fractal like finite grahs we consider related poincare constants and resistance and study their asymptotic behaviour we construct a markov semi group on fractals as a subsequence of random walks and study its properties finally we construct self similar diffusion processes on fractals which have a certain recurrence property and plenty of symmetries

nonlinear problems on the sierpinski gasket abstract this paper concerns with a class of elliptic equations on fractal domains depending on a real parameter our approach is based on variational methods more precisely the existence of at least two non trivial weak strong solutions for the treated problem is obtained exploiting a local minimum theorem for differentiable functionals defined on reflexive banach spaces a special case of the main result improves a classical application of the mountain pass theorem in the fractal setting given by falconer and hu in 14

weyl s problem for the spectral distribution of laplacians on p c f self similar fractals we establish an analogue of weyl s classical theorem for the asymptotics of eigenvalues of laplacians on a finitely ramified i e p c f self similar fractalk such as for example the sierpinski gasket we consider both dirichlet and neumann boundary conditions as well as laplacians associated with bernoulli type multifractal measures onk from a physical point of view we study the density of states for diffusions or for wave propagation in fractal media more precisely let x be the number of eigenvalues less thanx then we show that x is of the order ofxds 2 asx where the spectral exponent ds is computed in terms of the geometric as well as analytic structures ofk further we give an effective condition that guarantees the existence of the limit ofx ds 2 x asx this condition is in some sense generic in addition we define in terms of the above spectral exponents and calculate explicitly the spectral dimension ofk

transition densities for brownian motion on the sierpinski carpet upper and lower bounds are obtained for the transition densitiesp t x y of brownian motion on the sierpinski carpet these are of the same form as those which hold for the sierpinski gasket in addition the joint continuity ofp t x y is proved the existence of the spectral dimension is established and the einstein relation connecting the spectral dimension the hausdorff dimension and the resistance exponent is shown to hold