density estimates for a fluid jet model in the heisenberg group abstract in the heisenberg group setting we consider the fluid jet functional h n w 2 q w 1 d and we prove some density estimates for local minima in particular for large r the measure of the minimal jet in balls of radius r centered at the jet is of the order of r q 1 while the measure of the exterior medium is of the order of r q where q 2 n 1 is the homogeneous dimension also we obtain that h n u q in a suitable weak sense at the free boundary u 1

reduction of substructural interface degrees of freedom in flexibility based component mode synthesis a flexibility based component mode synthesis cms is proposed for reduced order modelling of dynamic behaviour of large structures the approach employs partitioning via the localized lagrange multiplier method the use of the localized lagrange multipliers leads to unlike the classical lagrange multipliers a linearly independent set of interface forces without any redundancies at multiply connected interface nodes the flexibility based cms method has shown significant advantages over the classical craig bampton method a key feature of the method is its substructural mode selection criterion that is independent of loading conditions unlike the majority of available cms approaches where one retains the full dimension of partition boundary degrees of freedom dofs the flexibility based method allows to reduce significantly the interface dofs the reduction of interface dofs represents the major contribution of the present communication the efficiency of the proposed approach is demonstrated on an analysis of a simple plate partitioned and of a more complex 3d structure both partitioned into several substructures copyright 2006 john wiley sons ltd

on uniformly subelliptic operators and stochastic area let x a be a markov process with generator i j i aij j where a is a uniformly elliptic symmetric matrix thanks to the fundamental works of t lyons stochastic differential equations driven by x a can be solved in the rough path sense that is pathwise by using a suitable stochastic area process our construction of the area which generalizes previous works of lyons stoica and then lejay is based on dirichlet forms associated to subellitpic operators this enables us in particular to discuss large deviations and support descriptions in suitable rough path topologies as typical rough path corollary freidlin wentzell theory and the stroock varadhan support theorem remain valid for stochastic differential equations driven by x a

weighted poincare inequality and heat kernel estimates for finite range jump processes it is well known that there is a deep interplay between analysis and probability theory for example for a markovian infinitesimal generator mathcal l the transition density function p t x y of the markov process associated with mathcal l if it exists is the fundamental solution or heat kernel of mathcal l a fundamental problem in analysis and in probability theory is to obtain sharp estimates of p t x y in this paper we consider a class of non local integro differential operators mathcal l on mathbb r d of the form mathcal l u x lim limits varepsilon downarrow 0 int limits y in mathbb r d y x varepsilon u y u x j x y dy where displaystyle j x y frac c x y x y d alpha bf 1 x y leq kappa for some constant kappa 0 and a measurable symmetric function c x y that is bounded between two positive constants associated with such a non local operator mathcal l is an mathbb r d valued symmetric jump process of finite range with jumping kernel j x y we establish sharp two sided heat kernel estimate and derive parabolic harnack principle for them along the way some new heat kernel estimates are obtained for more general finite range jump processes that were studied in barlow et al in trans am math soc 2008 one of our key tools is a new form of weighted poincare inequality of fractional order which corresponds to the one established by jerison in duke math j 53 2 503 523 1986 for differential operators using meyer s construction of adding new jumps we also obtain various a priori estimates such as holder continuity estimates for parabolic functions of jump processes not necessarily of finite range where only a very mild integrability condition is assumed for large jumps to establish these results we employ methods from both probability theory and analysis extensively

proper orthogonal decomposition with high number of linear constraints for aerodynamical shape optimization shape optimization involving finite element analysis in engineering design is frequently hindered by the prohibitive cost of function evaluations reduced order models based on proper orthogonal decomposition pod constitute an economical alternative however the truncation of the pod basis implies an error in the calculation of the global values used as objectives and constraints which in turn affects the optimization results in our former contribution xiao and breitkopf 2013 we have introduced a constrained pod projector allowing for exact linear constraint verification for a reduced order model nevertheless this approach was limited a to relatively low numbers constraints therefore in the present paper we propose an approach for a high number of constraints the main idea is to extend the snapshot pod by introducing a new constrained projector in order to reduce both the physical field and the constraint space this allows us to search for the pareto set of best compromises between the projection and the constraint verification errors thereby enabling fine tuning of the reduced model for a particular purpose we illustrate the proposed approach with the reduced order model of the flow around an airfoil parameterized with shape variables

balanced model reduction via the proper orthogonal decomposition a new method for performing a balanced reduction of a high order linear system is presented the technique combines the proper orthogonal decomposition and concepts from balanced realization theory the method of snapshotsisused to obtainlow rank reduced rangeapproximationsto thesystemcontrollability and observability grammiansineitherthetimeorfrequencydomain theapproximationsarethenusedtoobtainabalancedreducedorder model the method is particularly effective when a small number of outputs is of interest it is demonstrated for a linearized high order system that models unsteady motion of a two dimensional airfoil computation of the exact grammians would be impractical for such a large system for this problem very accurate reducedorder models are obtained that capture the required dynamics with just three states the new models exhibit far superiorperformancethanthosederived using a conventionalproperorthogonal decomposition although further development is necessary the concept also extends to nonlinear systems

topology of lattice gauge fields non abelian gauge fields on a four dimensional hypercubic lattice with small action density tr u dot p for su 2 gauge fields are shown to carry an integer topological chargeq which is invariant under continuous deformations of the field a concrete expression forq is given and it is verified thatq reduces to the familiar chern number in the classical continuum limit

a low cost goal oriented compact proper orthogonal decomposition basis for model reduction of static systems a novel model reduction technique for static systems is presented the method is developed using a goal oriented framework and it extends the concept of snapshots for proper orthogonal decomposition pod to include sensitivity derivatives of the state with respect to system input parameters the resulting reduced order model generates accurate approximations due to its goal oriented construction and the explicit training of the model for parameter changes the model is less computationally expensive to construct than typical pod approaches since efficient multiple right hand side solvers can be used to compute the sensitivity derivatives the effectiveness of the method is demonstrated on a parameterized aerospace structure problem

fundamental solutions for second order subelliptic operators estimation de la solution fondamentale d un operateur aux derivees partielles lineaire du second ordre sur une variete compacte avec une mesure reguliere

on semilinear δλ laplace equation abstract we prove some existence nonexistence and regularity results for the boundary value problem u f u 0 in u 0 where is a bounded subset of r n n 2 and is a laplacian i e a degenerate elliptic operator of the kind i 1 n x i i 2 x x i 1 n together with some assumptions made in franchi and lanconelli 1984 1 the family is supposed to verify a condition making homogeneous of degree two with respect to a group of dilations in r n

hardy littlewood theory for semigroups soit t t t 0 un semi groupe symetrique sousmarkovien et soit n 2 alors on a les proprietes equivalentes a dim t t n b il existe c 2 0 s t t t f c 2 t n 2 f 1 f l 1 x t 0 en fait a b meme dans le cas n 2

fine properties of functions with bounded variation in carnot caratheodory spaces abstract we study properties of functions with bounded variation in carnot caratheodory spaces we prove their almost everywhere approximate differentiability and we examine their approximate discontinuity set and the decomposition of their distributional derivatives under an additional assumption on the space called property r we show that almost all approximate discontinuities are of jump type and we study a representation formula for the jump part of the derivative

energy inequalities for cutoff functions and some applications let x d m be a metric measure space with a local regular dirichlet form we establish necessary and sufficient conditions for upper heat kernel bounds with subdiffusive space time exponent to hold this characterization is stable under rough isometries that is it is preserved under bounded perturbations of the dirichlet form further we give a criterion for stochastic completeness in terms of a sobolev inequality for cutoff functions as an example we show that this criterion applies to an anomalous diffusion on a geodesically incomplete fractal space where the well established criterion in terms of volume growth fails

isoperimetric and sobolev inequalities for carnot caratheodory spaces and the existence of minimal surfaces after hormander s fundamental paper on hypoellipticity 54 the study of partial differential equations arising from families of noncommuting vector fields has developed significantly in this paper we study some basic functional and geometric properties of general families of vector fields that include the hormander type as a special case similar to their classical counterparts such properties play an important role in the analysis of the relevant differential operators both linear and nonlinear to motivate our results we recall some classical inequalities let e c r be a caccioppoli set a measurable set having a locally finite perimeter then one has the isoperimetric inequality

a priori hyperreduction method an adaptive approach model reduction methods are usually based on preliminary computations to build the shape function of the reduced order model rom before the computation of the reduced state variables they are a posteriori approaches most of the time these preliminary computations are as complex as the simulation which we want to simplify by the rom the reduction method we propose avoids such preliminary computations it is an a priori approach based on the analysis of some state evolutions such that all the state evolutions needed to perform the model reduction are described by an approximate rom the rom and the state evolution are simultaneously improved by the method thanks to an adaptive strategy obviously an initial set of known shape functions can be used to define the rom to adapt but it is not necessary the adaptive procedure includes extensions of the subspace spanned by the shape functions of the rom and selections of the most relevant shape functions in order to represent the state evolution the hyperreduction is achieved by selecting a part of the integration points of the finite element model to forecast the evolution of the reduced state variables hence both the number of degrees of freedom and the number of integration points are reduced to perform the adaptive procedure different computational strategies can be developed in this paper we propose an incremental algorithm involving adaptive periods during these adaptive periods the incremental computation is restarted until a quality criterion is satisfied this approach is compatible with classical formulations of the equations

on poincare type inequalities using estimates of the heat kernel we prove a poincare inequality for star shape domains on a complete manifold the method also gives a lower bound for the gap of the first two neumann eigenvalues of a schrodinger operator

liouville theorem for x elliptic operators abstract we prove a liouville type theorem for a class of degenerate elliptic operators of the form l u i j 1 n x i a i j x j u i 1 n b i x i u l is supposed to be x elliptic with respect to a family x x 1 x m of locally lipschitz continuous vector fields in the sense introduced in e lanconelli a e kogoj x elliptic operators and x control distances contributions in honor of the memory of ennio de giorgi ricerche di matematica 49 suppl 2000 223 243

a subelliptic analogue of aronson serrin s harnack inequality we study the harnack inequality for weak solutions of a class of degenerate parabolic quasilinear pde begin aligned partial t u x i a i x t u xu b x t u xu end aligned in cylinders omega times 0 t where omega subset m is an open subset of a manifold m endowed with control metric d corresponding to a system of lipschitz continuous vector fields x x 1 ldots x m and a measure d sigma we show that the harnack inequality follows from the basic hypothesis of doubling condition and a weak poincare inequality in the metric measure space m d d sigma we also show that such hypothesis hold for a class of riemannian metrics g epsilon collapsing to a sub riemannian metric lim epsilon rightarrow 0 g epsilon g 0 uniformly in the parameter epsilon ge 0

constrained proper orthogonal decomposition based on qr factorization for aerodynamical shape optimization while performing numerical simulations for design optimization one of the major issues is reaching a good compromise between accuracy and computational effort simulation methods such as finite elements finite volumes etc provide high fidelity numerical solutions at a cost which may be prohibitive in optimization problems requiring frequent calls to the discretized equations solver the proper orthogonal decomposition pod constitutes an economical and efficient option to decrease the cost of the solution however the truncation of the pod basis implies an error in the calculation of the global quantities used as objectives and optimization constraints which in turn might bias the optimization results our idea is thus to improve the snapshot pod by means of a reduced basis constructed to provide an exact interpolation of the quantities of interest obtained by integration of the physical fields to this end we reformulate the pod as a minimization problem where the desired properties are expressed as a set of constraints impacting the calculation of both the modes and the coefficients the main contribution of this paper is to provide a detailed mathematical justification for our constrained variant of the pod including a graphical interpretation of the proposed approach the constrained pod method is then applied to the problem of representing the pressure field around a 2d wing and is compared with the traditional pod

non local dirichlet forms and symmetric jump processes we consider the non local symmetric dirichlet form e f given by with f the closure with respect to e 1 of the set of c 1 functions on r d with compact support where e 1 f f e f f f rd f x 2 dx and where the jump kernel j satisfies for 0 2 x y 1 this assumption allows the corresponding jump process to have jump intensities whose sizes depend on the position of the process and the direction of the jump we prove upper and lower estimates on the heat kernel we construct a strong markov process corresponding to e f we prove a parabolic harnack inequality for non negative functions that solve the heat equation with respect to e finally we construct an example where the corresponding harmonic functions need not be continuous

an alternative approach to the holder continuity of solutions to some elliptic equations abstract we give an alternative and self contained proof of the holder continuity of locally bounded solutions to some elliptic equations including the equation associated with the p laplacian operator we combine the techniques introduced by de giorgi and moser with a 1 dimensional poincare inequality

non linearly stable reduced order models for incompressible flow with energy conserving finite volume methods abstract a novel reduced order model rom formulation for incompressible flows is presented with the key property that it exhibits non linearly stability independent of the mesh of the full order model the time step the viscosity and the number of modes the two essential elements to non linear stability are 1 first discretise the full order model and then project the discretised equations and 2 use spatial and temporal discretisation schemes for the full order model that are globally energy conserving in the limit of vanishing viscosity for this purpose as full order model a staggered grid finite volume method in conjunction with an implicit runge kutta method is employed in addition a constrained singular value decomposition is employed which enforces global momentum conservation the resulting velocity only rom is thus globally conserving mass momentum and kinetic energy for non homogeneous boundary conditions a one time poisson equation is solved that accounts for the boundary contribution the stability of the proposed rom is demonstrated in several test cases furthermore it is shown that explicit runge kutta methods can be used as a practical alternative to implicit time integration at a slight loss in energy conservation

spectral gap estimates on compact manifolds for a compact riemannian manifold with boundary its mass gap is the difference between the first and second smallest dirichlet eigenvalues in this paper taking a variational approach we obtain an explicit lower bound estimate of the mass gap for any compact manifold in terms of geometric quantities 0 introduction let m g be a compact riemannian manifold with non empty boundary m the laplace beltrami operator laplacian acting on functions on m is defined by f 1 g i gg f j where the metric on m is given by ds gijdx dx g gij 1 and g det gij eigenvalues of the laplacian under dirichlet boundary conditions are constants li which satisfy ui liui 0 x m ui x 0 x m for some nonzero eigenfunctions ui it is well known that the spectrum of the dirichlet problem satisfies specd m 0 l1 l2 l3 so that the gap l2 l1 sometimes referred to as the mass gap is nontrivial in s w y y the mass gap is estimated by analyzing the function u2 u1 they show that 2 log u1 log l2 l1 0 therefore l2 l1 appears as an eigenvalue of a certain partial differential operator moreover since it is easy to see that is smooth on m and satisfies 0 l2 l1 is a neumann eigenvalue of a certain partial differential operator thus the gradient estimate techniques of li and yau l y can be employed to show that l2 l1 4d2 where m is a convex euclidean domain and d is the diameter of m the assumption of convexity seems to be crucial however to the argument since the hessian received by the editors august 22 1995 and in revised form february 13 1997 1991 mathematics subject classification primary 58c40 c 1999 american mathematical society

gaussian bounds and parabolic harnack inequality on locally irregular graphs a well known theorem of delmotte is that gaussian bounds parabolic harnack inequality and the combination of volume doubling and poincare inequality are equivalent for graphs in this paper we consider graphs for which these conditions hold but only for sufficiently large balls and prove a similar equivalence

long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator abstract second order subelliptic operators have been the subject of a considerable amount of research in recent years starting with the paper rothschild and stein in which the sharp form of hormander s famous subellipticity theorem is proved and continuing through the work of fefferman and phong and sanchez calle it has become increasingly clear that precise regularity estimates for these operators depend intimately on the geometry associated with the operator under consideration the main purpose of this article is to obtain bounds from above and below on p t x y t epsilon 1 infinity in terms of standard heat kernels

potential analysis for a class of diffusion equations a gaussian bounds approach let h be a linear second order partial differential operator with non negative characteristic form in a strip s r n r we assume that h as a fundamental solution smooth out of its poles and bounded from above and from below by gaussian kernels modeled on subriemannian doubling distances in r n under these assumptions we show that h endows s with a structure of harmonic space this allows us to study boundary value problems for l with a perron wiener brelot bauer method and to obtain pointwise regularity estimates at the boundary in terms of wiener series modeled on the gaussian kernels our analysis includes the proof of a scale invariant harnack inequality for nonnegative solutions we also show an application to the real hypersurphaces of c n 1 with given levi curvature

existence of isoperimetric regions in contact sub riemannian manifolds abstract we prove existence of regions minimizing perimeter under a volume constraint in contact sub riemannian manifolds whose quotient by the group of contact transformations preserving the sub riemannian metric is compact

attractors met x elliptic operators abstract we consider degenerate parabolic and damped hyperbolic equations involving an operator l that is x elliptic with respect to a family of locally lipschitz continuous vector fields x x 1 x m the local well posedness is established under subcritical growth restrictions on the nonlinearity f which are determined by the geometry and functional setting naturally associated with the family of vector fields x assuming additionally that f is dissipative the global existence of solutions follows and we can characterize their longtime behavior using methods from the theory of infinite dimensional dynamical systems

analysis on lie groups theorie locale diffusion de la chaleur sur des groupes de lie connexes la theorie globale les inegalites de sobolev bases graduees sur des corps lisses construction des bases graduees estimation locale du noyau diffusion de la chaleur demonstration de l inegalite de sobolev locale formes de dirichlet sur des groupes le semi groupe de convolution les conditions de sobolev modifiees

harnack inequality for non divergence form operators on stratified groups we prove lower bounds for the fundamental solutions of the non divergence form operators j i j x t x i x j t and i j i j x x i x j where the x i s are hormander vector fields generating a stratified group g and a i j i j is a positive definite matrix with holder continuous entries we then prove an invariant harnack inequality for such operators as a byproduct we also study some relevant properties of the green functions on bounded domains

heat kernel based decomposition of spaces of distributions in the framework of dirichlet spaces classical and nonclassical besov and triebel lizorkin spaces with complete range of indices are developed in the general setting of dirichlet space with a doubling measure and local scale invariant poincare inequality this leads to a heat kernel with small time gaussian bounds and holder continuity which play a central role in this article frames with band limited elements of sub exponential space localization are developed and frame and heat kernel characterizations of besov and triebel lizorkin spaces are established this theory in particular allows the development of besov and triebel lizorkin spaces and their frame and heat kernel characterization in the context of lie groups riemannian manifolds and other settings

smoothness of subriemannian isometries we show that the group of isometries i e distance preserving homeomorphisms of an equiregular subriemannian manifold is a finite dimensional lie group of smooth transformations the proof is based on a new pde argument in the spirit of harmonic coordinates establishing that in an arbitrary subriemannian manifold there exists an open dense subset where all isometries are smooth

new integral estimates for deformations in terms of their nonlinear strains if u is a bi lipschitzian deformation of a bounded lipschitz domain in ln n 2 we show that the lp norm p 1 p n of a certain nonlinear strain function e u associated with u dominates the distance in lq q np n p if p n from u to a suitably chosen rigid motion of n this work extends that of f john who proved corresponding estimates for p 1 under the hypothesis that u has uniformly small strain we also obtain a bound for the oscillation of du in l2 these estimates are apparently the first to apply with no a priori pointwise hypotheses upon the strain of u in 3 the integral int limits omega e u 2d 3 is dominated by typical hyperelastic energy functionals proposed in the literature for modeling the behavior of rubber thus the case n 3 p 2 gives the first general bound for the deformations of such materials in terms of the associated nonlinear elastic work

the computation of characteristic classes of lattice gauge fields agl p c valued lattice gauge fieldu on a simplicial complex determines a principalgl p c bundle if the plaquette products are sufficiently small with respect to the maximum distortion coefficient of the transporters a representative cocyclecq for theqth chern class of can be computed on each 2q simplex by takingcq to be the intersection number of a certain singular 2q cubem with a schubert type variety q in the space of allp p matrices this reduces to the solution of polynomial equations with coefficients coming fromu and thus avoids numerical integration or cooling type procedures an application of this method is suggested for the computation of the topological charge of ansu 3 valued lattice gauge field on a 4 complex

locally strongly convex affine hyperspheres realizing chen s equality abstract in affine differential geometry of hypersurface c scharlach et al found an inequality involving intrinsic and extrinsic curvatures and classified elliptic and hyperbolic affine hyperspheres realizing the equality if an affine invariant 2 dimensional distribution d 2 is integrable in this paper we continue to study affine hyperspheres realizing the equality including parabolic affine hyperspheres as main results firstly we classify parabolic affine hyperspheres realizing the equality if its scalar curvature is constant or d 2 is integrable next by introducing a well defined 3 dimensional distribution d 3 when d 2 is not integrable we complete the classification of locally strongly convex affine hyperspheres realizing the equality if d 3 is integrable finally we pose a conjecture and a problem in order to determine all affine hyperspheres attaining the equality

explicit constants for gaussian upper bounds on heat kernels on utilise des inegalites logarithmiques de sobolev pour obtenir des bornes superieures gaussiennes sur les noyaux de la chaleur associes a des operateurs d ordre 2 elliptiques et hypoelliptiques

stein weiss inequalities for the fractional integral operators in carnot groups and applications in this article we consider the fractional integral operator i on any carnot group x1d53e i e nilpotent stratified lie group in the weighted lebesgue spaces l p x x1d53e we establish stein weiss inequalities for i and obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional integral operator i from the spaces l p x x1d53e to l q x x1d53e and from the spaces l 1 x x1d53e to the weak spaces wl q x x1d53e by using the stein weiss inequalities in the limiting case we prove that the modified fractional integral operator is bounded from the space l p x x1d53e to the weighted bounded mean oscillation bmo space bmo x x1d53e where q is the homogeneous dimension of x1d53e as applications of the properties of the fundamental solution of sub laplacian on x1d53e we prove two sobolev stein embedding theorems on weighted lebesgue and weighted besov spaces in the carnot group setting as another application we prove the boundedness of i from the weighted besov

boundedness and regularity of solutions of degenerate elliptic partial differential equations abstract we study quasilinear degenerate equations with their associated operators satisfying sobolev and poincare inequalities our problems are similar to those in sawyer and wheeden 2006 20 but we consider more general integral equations on abstract sobolev spaces indeed we provide a unified approach via abstract setting our main tool is moser iteration we obtain boundedness assuming a weighted sobolev embedding of weak solutions and also harnack inequalities for nonnegative solutions assuming further that a weighted poincare inequality holds for their logarithm and existence of a sequence of cutoff functions and hence holder s continuity of solutions our assumption is weaker than the above mentioned paper and other existing papers even in the case where the setting is on r n for example we do not assume any doubling condition and we are able to deal with weighted estimates our results are new even when the leading term in the equations are vector fields that satisfy hormander s conditions

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

quasiregular mappings of the heisenberg group the condition 1 1 means that for all d g the coordinate functions o f f belong to the sobolev space w d of functions in l d whose distributional first order partial derivatives are also l integrable in d in the above definition f x is the formal derivative of f a t x defined by means of the partial derivatives i f x l is the operator norm of f x and ji x is the jacobian determinant of f a t x these are uniquely defined almost everywhere by 1 1 it turns out that quasiregular mappings are differentiable almost everywhere and a nonconstant quasiregular map is discrete open and sense preserving for these and other basic properties of qr maps we refer to books re by reshetnyak vu by vuorinen and also to a forthcoming book r6 by the second author the definition extends immediately to the case f m n where m and n are connected oriented riemannian n manifolds we say that a continuous map f m n is quasiregular if it has a local representation at each point of m which is quasiregular in the above sense and if there exists k 1 k oe such that

optimal partial regularity for weak solutions of nonlinear sub elliptic systems in carnot groups abstract this paper is concerned with partial regularity for weak solutions to nonlinear sub elliptic systems in divergence form in carnot groups the technique of a harmonic approximation introduced by simon and developed by duzaar and grotowski is adapted to our context we establish caccioppoli type inequalities and partial regularity with optimal local holder exponents for horizontal gradients of weak solutions to systems under super quadratic natural structure conditions and super quadratic controllable structure conditions respectively

operateurs uniformement sous elliptiques sur les groupes de lie given a connected unimodular lie group g having polynomial volume growth and left invariant vector fields x1 xk which generate the lie algebra of g consider the operator la i j 1kxiaijxj where a g rk rk is a smooth symmetric matrix valued function which satisfies a 2 i j 1k aij x i j 1 2 x g rk for some 0 1 what we show is that the corresponding heat flow semigroup etla admits a kernel which satisfies two sided gaussian estimates in terms of the control distance determined by the xi s moreover the estimates can be made to depend on a only through we also prove harnack inequalities and holder regularity of solutions we end with a discussion of some other sub elliptic situations

green functions for weighted subelliptic p laplace operator constructed by hormander s vector fields abstract this paper deals with the weighted subelliptic p laplace operator constructed by hormander s vector fields l p u d i v x a x x u x x u x p 2 2 a x x u x where u w 1 p u w 1 p q and a x is a bounded measurable and m m symmetric matrix satisfying 1 w x 2 p 2 a x w x 2 p 2 r m w x a p we first prove existence of the modified green function for l p by virtue of minty browder theorem and then existence of the green function for l p by checking the convergence of sequence of modified green functions next we derive upper bounds of the modified green function for l p by establishing the interpolation inequality in the weighted weak l p spaces finally the bounds of the green function for l p are also obtained by virtue of results for the modified green function and the compact weighted embedding theorem

the weighted isoperimetric type and sobolev type inequalities for hypersurfaces in carnot groups abstract this paper mainly studies hypersurfaces immersed in a sub riemannian step k carnot group g with a density e f where f is a smooth function on g we give the first variation formula about the weighted horizontal area of hypersurfaces furthermore on the base of some work we derive the weighted isoperimetric type and sobolev type inequalities for hypersurfaces

harnack inequalities and gaussian estimates for a class of hypoelliptic operators we prove a global harnack inequality for a class of degenerate evolution operators by using repeatedly an invariant local harnack inequality as a consequence we obtain an accurate gaussian lower bound for the fundamental solution for some meaningful families of degenerate operators

lattice gauge fields principal bundles and the calculation of topological charge su2 valued lattice gauge fields are studied on a 4 dimensional simplicial lattice ifu has sufficiently small plaquette products then there is a unique principal su2 bundle admitting transition functions defined on the intersections of adjacent dual cells which take values within 8 ofu an algorithm is explicitly given which associates an integer to everyu off a certain set of measure zero this algorithm only involves evaluation of 4 4 determinants and the solution of quadratic equations whenu is as above the integer produced is the second chern number of i e the topological charge ofu

pointwise local estimates and gaussian upper bounds for a class of uniformly subelliptic ultraparabolic operators abstract we consider a class of second order ultraparabolic differential equations in the form t u i j 1 m x i a i j x j u x 0 u where a a i j is a bounded symmetric and uniformly positive matrix with measurable coefficients under the assumption that the operator i 1 m x i 2 x 0 t is hypoelliptic and the vector fields x 1 x m and x 0 t are invariant with respect to a suitable homogeneous lie group we adapt the moser s iterative methods to the non euclidean geometry of the lie groups and we prove an l loc bound of the solution u in terms of its l loc p norm we then use a technique going back to aronson to prove a pointwise upper bound of the fundamental solution of the operator i j 1 m x i a i j x j x 0 t the bound is given in terms of the value function of an optimal control problem related to the vector fields x 1 x m and x 0 t finally by using the upper bound the existence of a fundamental solution is then established for smooth coefficients a i j

parabolic harnack inequality and heat kernel estimates for random walks with long range jumps we investigate the relationships between the parabolic harnack inequality heat kernel estimates some geometric conditions and some analytic conditions for random walks with long range jumps unlike the case of diffusion processes the parabolic harnack inequality does not in general imply the corresponding heat kernel estimates

a family of diffusion processes on sierpinski carpets we construct a family of diffusions p px on the d dimensional sierpinski carpet f the parameter ranges over d h where d h log 3 d 1 log 3 is the hausdorff dimension of the d dimensional sierpinski carpet f these diffusions p are reversible with invariant measures here are radon measures whose topological supports are equal to f and satisfy self similarity in the sense that 3a 3 a for all a f in addition the diffusion is self similar and invariant under local weak translations cell translations of the sierpinski carpet the transition density p p t x y is locally uniformly positive and satisfies a global gaussian upper bound in spite of these well behaved properties the diffusions are different from barlow bass brownian motions on the sierpinski carpet