similarities of the integral pade approximants ii in this paper we show further similarities of the integral pade approximant the similarity of the integral pade approximant with the functional pade approximant and the improved functional epsilon algorithms is demonstrated the purpose of demonstrating the similarity between the integral pade approximant and the functional pade approximant is to verify our conjecture which is the integral pade approximant and the hybrid functional pade approximant produce identical rational functions finally we demonstrate the improvement of the classical functional epsilon algorithms which also has a relationship with the integral pade approximant

uniform boundary harnack principle and generalized triangle property let d be a bounded open subset in rd d 2 and let g denote the green function for d with respect to 2 0 2 d if 2 assume that d satisfies the interior corkscrew condition if 2 i e if g is the classical green function on d assume more restrictively that d is a uniform domain let g g y0 1 for some y0 d based on the uniform boundary harnack principle it is shown that g has the generalized triangle property which states that g z y g z cg x y g x when d z x d z y an intermediate step is the approximation g x y x y dg x g y g a 2 where a is an arbitrary point in a certain set b x y this is discussed in a general setting where d is a dense open subset of a compact metric space satisfying the interior corkscrew condition and g is a quasi symmetric positive numerical function on d d which has locally polynomial decay and satisfies harnack s inequality under these assumptions the uniform boundary harnack principle the approximation for g and the generalized triangle property turn out to be equivalent

uniform boundedness of conditional gauge and schrodinger equations we prove that for a bounded domaind rn withc2 boundary and q in k n loc n geqq 3 if e x exp int limits 0 tau d q x t dt mathop ddag limits infty ind then mathop sup limits mathop x in d limits z in partial d e z x exp int limits 0 tau d q x t dt infty xt brownian motion

conditional gauge and potential theory for the schrodinger operator on etend le theoreme de jauge conditionnelle a des operateurs plus generaux et a des domaines moins reguliers on obtient des resultats de theorie du potentiel pour l equation de schrodinger

intrinsic ultracontractivity and eigenfunction estimates for schrodinger operators let d be a domain in rn n 2 and let h h0 v where h0 is a divergence form uniformly elliptic operator with dirichlet boundary conditions and v is in the kato class intrinsic ultracontractivity is proved for the semigroup of h when d is holder domain of order 0 or a uniformly holder domain of order for 0 2 for every 2 there exists a uniformly holder domain of order for which the dirichlet laplacian is not intrinsically ultracontractive for a large class of domains it is shown that the heat kernels for h0 and h decay at the same rate at the boundary applications are given to the lifetime of conditioned brownian motion some of our results seem to be new even for smooth domains

function valued pade type approximant via the formal orthogonal polynomials and its applications in solving integral equations a kind of function valued pade type approximant via the formal orthogonal polynomials fptavop is introduced on the polynomial space and an algorithm is sketched by means of the formal orthogonal polynomials this method can be applied to approximate characteristic values and the corresponding characteristic function of fredholm integral equation of the second kind moreover theoretical analyses show that fptavop method is the most effective one for accelerating the convergence of a sequence of functions in addition a typical numerical example is presented to illustrate when the estimates of characteristic value and characteristic function by using this new method are more accurate than other methods

existence and uniqueness of weak solutions for a non uniformly parabolic equation in this paper we develop a unifying method to prove the existence and uniqueness of weak solutions for the initial boundary value problem of a non uniformly parabolic equation some well known parabolic equations are the special cases of this equation

singular solutions for semi linear parabolic equations on nonsmooth domains we prove the existence of positive singular solutions for the semi linear parabolic equation xu tu up 0 on d 0 where p 1 d is a bounded nta domain in rn n 2 and is in a general class of signed radon measures on d covering the elliptic kato class of potentials adopted by zhang and zhao a new proof of the result based on a simple fixed point theorem is also given

extrapolation methods for vector sequences this is an expository paper that describes and compares five methods for extrapolating to the limit or anti limit of a vector sequence without explicit knowledge of the sequence generator the methods are the minimal polynomial extrapolation mpe studied by cabay and jackson mesina and skelboe the reduced rank extrapolation rre of eddy which we show to be equivalent to mesina s version of mpe the vector and scalar versions of the epsilon algorithm vea sea introduced by wynn and extended by brezinski and gekeler and the topological epsilon algorithm tea of brezinski we cover the derivation and error analysis of iterated versions of the algorithms as applied to both linear and nonlinear problems and we show why these versions tend to converge quadratically we also present samples from extensive numerical testing that has led us to the following conclusions a tea in spite of its role as a theoretical link between the polynomial type and the epsilon type methods has no practical appl

gaugeability and conditional gaugeability new kato classes are introduced for general transient borel right processes for which gauge and conditional gauge theorems hold these new classes are the genuine extensions of the green tight measures in the classical brownian motion case however the main focus of this paper is on establishing various equivalent conditions and consequences of gaugeability and conditional gaugeability we show that gaugeability conditional gaugeability and the subcriticality for the associated schrodinger operators are equivalent for transient borel right processes with strong duals analytic characterizations of gaugeability and conditional gaugeability are given for general symmetric markov processes these analytic characterizations are very useful in determining whether a process perturbed by a potential is gaugeable or conditionally gaugeable in concrete cases connections with the positivity of the spectral radii of the associated schrodinger operators are also established

nonlinear elliptic systems with general growth we prove local lipschitz continuity and as a consequence c k and c regularity of weak solutions u for a class of nonlinear elliptic differential systems of the formn 1 xi a i du 0 1 2 m the growth conditions on the dependence of functions a i on the gradient du are so mild to allow us to embrace growths between the linear and the exponential cases and they are more general than those known in the literature

positive harmonic functions on complete manifolds of negative curvature this paper studies the interaction between the geometry of complete riemannian manifolds of negative curvature and some aspects of function theory on these spaces the study of harmonic functions on the unit disc provides a classical and beautiful example of this interaction we recall some aspects of this below there is a well known representation of positive harmonic functions on the unit disc u due to herglotz 13 in terms of positive borel measures tt on the circle s1

row convergence theorems for generalised inverse vector valued pade approximants abstract the approximants mentioned in the title are related to vector valued continued fractions and the vector algorithm devised by wynn in 1963 here we establish a unitary invariance property of these approximants and describe how the classical 1 dimensional pade approximants can be obtained as a special case the main results of the paper consist of de montessus de ballore type convergence theorems for row sequences having fixed denominator degree of vector valued approximants to meromorphic vector functions

conditional gauge with unbounded potential let b be a ball in r d and q be a borel function on it we prove that if mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9gqpe0 rq0dxdd9vqaq jfrvkfhe9pgea0dxdar jb9hs0dxdbpyxe9vr0 vr 0 vqpwqaaeaabigaciaacaqabeaadaqaaqaaaoqaamaaxababagaci 4caiaacwhacaggwbaaleaacawg4bgaeyici4sabmoqayaaraaabeaa ieaakiaa bcadawdrbqaamaalaaabagaaiifaiaadghacaggoagaam yeaiaacmcacagg8baabagaaiifaiaadihacqghsislcawg5bgaaiif amaacaaaleqabagaamizaiabgkhitiaaikdaaaaaaogaa8hiagqaci aa rgacagf5baaleaacawgcbaabeqdcqghrii8aaaa 4fb6 mathop sup limits x in bar b int limits b frac q y x y d 2 dy is sma11 enough then mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9gqpe0 rq0dxdd9vqaq jfrvkfhe9pgea0dxdar jb9hs0dxdbpyxe9vr0 vr 0 vqpwqaaeaabigaciaacaqabeaadaqaaqaaaoqaamaaxababagaci 4caiaacwhacaggwbaaleaadawfqaqaaiaadihacqghiiizcawgcbaa meaacawg6bgaeyici4saeyoaiyraamoqaaqabaaaleqaagqaaogaa8 hiagqaciaa veadaqhaawcbagaa4neaaqaaiaa hhaaagcdawadaqa aigacwgacagg4bgaaiicaiaa bcadawdxbqaaiaa fhacawfoagaa4 heamaabaaaleaacagf0baabeaakiaacmcacawgkbgaamidaawcbaga a8hmaaqaaiaadshadawgaaadbagaamoqaaqabaaaniabguiiydaaki aawufacagldbaacawfgagaa8hpaiaa bcacawfragaa8hiaiabg6hi lcaa 5a6b mathop sup limits mathop x in b limits z in partial b e z x left exp int limits 0 t b q x t dt right infty this paper gives a new proof of one of the two main results by aizenman and simon in 1 by a simple and elementary method a basic theorem in chung and rao 2 is extended to the class of q treated in 1

nonlinear wave equations as limits of convex minimization problems proof of a conjecture by de giorgi we prove a conjecture by de giorgi which states that global weak solutions of nonlinear wave equations such as w jwj p 2 w 0 can be obtained as limits of functions that minimize suitable functionals of the calculus of variations these functionals which are integrals in space time of a convex lagrangian contain an exponential weight with a parameter and the initial data of the wave equation serve as boundary conditions as tends to zero the minimizers v converge up to subsequences to a solution of the nonlinear wave equation there is no restriction on the nonlinearity exponent and the method is easily extended to more general equations

estimates on green functions and poisson kernels for symmetric stable processes one of the most basic and most important subfamily of levy processes is symmetric stable processes a symmetric stable process x on rn is a levy process whose transition density p t x y relative to the lebesgue measure is uniquely determined by its fourier transform rn e ix p t x dx e t here must be in the interval 0 2 when 2 we get a brownian motion running with a time clock twice as fast as the standard one brownian motion plays a central role in modern probability theory and has numerous important applications in other scientific areas as well as in many other branches of mathematics thus it has been intensively studied in this paper symmetric stable processes are referred to the case when 0 2 unless otherwise specified in the last few years there has been an explosive growth in the study of physical and economic systems that can be successfully modeled with the use of stable processes stable processes are now widely used in physics operations research queuing theory mathematical finance and risk estimation in some physics literatures symmetric stable processes are called levy flights and they have been applied to a wide range of very complex physics issues such as turbulent diffusion vortex dynamics anomalous diffusion in rotating flows and molecular spectral fluctuations in mathematical finance stable processes can be used to model stock returns in incomplete market for these and more applications of stable processes please see the interesting book 14 by janicki and weron and the references therein and the recent article 15 by klafter shlesinger and zuomofen in order to make precise predictions about natural phenomena and to better cope with these

local boundedness of weak solutions to the diffusive wave approximation of the shallow water equations abstract in this paper we prove that weak solutions to the diffusive wave approximation of the shallow water equations t u u z u 1 u f are locally bounded here u describes the height of the water z is a given function that represents the land elevation and f is a source term accounting for evaporation infiltration or rainfall

subcriticality and gaugeability of the schrodinger operator we investigate a schrodinger operator 2 v in r d d 3 with a potential v in the class k d satisfying a similar kato condition at infinity and prove an equivalence theorem connecting various conditions on subcriticality strong positivity and gaugeability of the operator

an existence result for dissipative nonhomogeneous hyperbolic equations via a minimization approach abstract we discuss a purely variational approach to the study of a wide class of second order nonhomogeneous dissipative hyperbolic pdes precisely we focus on the wave like equations that present also a nonzero source term and a first order in time linear term the paper carries on the research program initiated in 14 and developed in 15 21 on the de giorgi approach to hyperbolic equations

the martin kernel and infima of positive harmonic functions let d be a bounded lipschitz domain in bbr n and let k x z x in d z in partial d be the martin kernel based at x 0 in d for x y in d let k x y inf h x h mbox positive harmonic in d h y 1 we show that the function k completely determines the family of positive harmonic functions on d precisely for every z in partial d lim y rightarrow z k x y k x 0 y k x z the same result is true for second order uniformly elliptic operators and schr o dinger operators

positivity for equations involving polyharmonic operators with dirichlet boundary conditions cranston fabes and zhao 26 5 established the uniform bound sup x y 2 x 6 y r g1 n x z g1 n z y dz g1 n x y m 1 1 where g1 n x y is the green function for the laplacian with dirichlet boundary conditions on a lipschitz domain rn with n 3 see 27 for n 2 this estimate was used in 23 and 18 to obtain positivity uniformly with respect to f 0 for noncooperative elliptic systems as

a singular solution with smooth initial data for a semilinear parabolic equation abstract we consider the cauchy problem for a parabolic partial differential equation with a power nonlinearity our concern in this paper is the existence of a singular solution with smooth initial data by using the haraux weissler equation it is shown that there exist singular forward self similar solutions using this result we also obtain a sufficient condition for the singular solution with general initial data including smooth initial data

introduction to the improved levin type algorithms for accelerating convergence of sequence we describe two new algorithms for accelerating the convergence of scalar sequence we express these new algorithms as levin transformation the definitions of these new algorithms namely the improved levin algorithm and the modified levin algorithm are given we examine the effectiveness of the improved levin algorithm and the modified levin algorithm together with the well established methods namely wynn s r algorithm and brezinski s q algorithm for approximating the partial sum of a given power series estimates of the partial sum produced by the improved levin algorithm are found to be substantially more accurate than the original levin transformation and the wynn s r algorithm and it is similar to the modified levin algorithm and the brezinski s q algorithm

problems and progress in vector pade approximation abstract some open problems in vector pade approximation are stated and some recent de montessus type theorems governing convergence of rows of the vector pade table are contrasted weshow how these results indicate when it is more appropriate to use generalised inverse vector pade approximants or their hybridised form we show how a straightforward analogue of the berlekamp massey algorithm may be used to calculate generalised inverse vector pade approximants this algorithm is applied to the derivation of a new low order hybrid vector approximant related results include the case of a row convergence theorem for a complex valued power series using a clifford inverse instead of the moore penrose inverse

existence of parabolic minimizers on metric measure spaces abstract the objects of our studies are vector valued parabolic minimizers u associated to a convex caratheodory integrand f obeying a p growth assumption from below and a certain monotonicity condition in the gradient variable here the functions being considered are defined on a metric measure space x d for such parabolic minimizers that coincide with a time independent cauchy dirichlet datum u 0 on the parabolic boundary of a space time cylinder 0 t with an open subset x and t 0 we prove existence in the parabolic newtonian space l p 0 t n 1 p r n in this paper we generalize results from bogelein et al 2014 2015 to the metric setting and argue completely on a variational level

convergence and stability analyses for some vector extrapolation methods on the presence of defective iteration matrices abstract in two previous papers 10 11 convergence and stability results for the following vector extrapolation methods were presented minimal polynomial extrapolation reduced rank extrapolation modified minimal polynomial extrapolation and topological epsilon algorithm the analyses were carried out for vector sequences that include those arising from iterative methods for linear systems of equations having diagonalizable iteration matrices in this paper the analyses of 10 11 are extended to vector sequences that include those arising from iterative methods for linear systems having defective iteration matrices the results are illustrated with numerical examples the analyses above naturally suggest some old and some new extensions of the well known power method enabling one to obtain estimates for several dominant eigenvalues of a general matrix

time dependent singularities in semilinear parabolic equations existence of solutions abstract we study the existence of solutions of the semilinear parabolic equation t u u u p in d r n i here n 2 p 1 n 2 p n and i r is an open interval it is assumed that d r n i is supported on t t r n 1 t i for some c 1 2 i r n and is given by i t t d t where is a radon measure on i satisfying a b b a a b i for some positive function in the case where i is bounded we give conditions on the behavior of near 0 for the existence and nonexistence of solutions the existence for i r is also shown under additional conditions on p and the behavior of near moreover the singularity of solutions at x t is examined

estimates of heat kernel of fractional laplacian perturbed by gradient operators we construct a continuous transition density of the semigroup generated by delta alpha 2 b x cdot nabla for 1 alpha 2 d ge 1 and b in the kato class mathcal k d alpha 1 on mathbb r d for small time the transition density is comparable with that of the fractional laplacian

extrapolation methods for vector sequences an analogue of aitken s 2 method suitable for vector sequences is proposed aspects of the numerical performance of the vector algorithm based on using the moore penrose inverse are investigated the fact that the denominator polynomial associated with a vector pade approximant is the square of its equivalent in the scalar case is shown to be a source of approximation error in cases where the convergence of the vector sequence is dominated by real eigenvalues a hybrid form of the vector pade approximant having a denominator polynomial of minimal degree is proposed and its effectiveness is demonstrated on several standard examples

on time adaptive critical variable exponent vectorial diffusion flows and their applications in image processing i analysis abstract variable exponent spaces have found interesting applications in real world problems in imaging models the variable exponent can approach the critical value 1 and this poses unique challenges in proving existence of solutions for the corresponding pdes in this work we develop some new functional framework to study time dependent parabolic variable exponent flows specifically we consider bounded vectorial partial variation b v p v space and its variable exponent counterpart we then prove the existence of weak solutions to the critical vectorial p t x laplacian flow in the variable exponent b v p v space for time independent critical vectorial p x laplacian flow we obtain a unique semigroup solution our results are in particular valid in the scalar case and solve a long standing open problem

existence of evolutionary variational solutions via the calculus of variations abstract in this paper we introduce a purely variational approach to time dependent problems yielding the existence of global parabolic minimizers that is 0 t u t f x d u d x d t 0 t f x d u d d x d t whenever t 0 and c 0 0 t r n for the integrand f r n n 0 we merely assume convexity with respect to the gradient variable and coercivity these evolutionary variational solutions are obtained as limits of maps depending on space and time minimizing certain convex variational functionals in the simplest situation with some growth conditions on f the method provides the existence of global weak solutions to cauchy dirichlet problems of parabolic systems of the type t u div d f x d u 0 in 0

introduction to the improved functional epsilon algorithm this paper introduces the improved functional epsilon algorithm we have defined this new method in principle of the modified aitken 2 algorithm moreover we have found that the improved functional epsilon algorithm has remarkable precision of the approximation of the exact solution and there exists a relationship with the integral pade approximant the use of the improved functional epsilon algorithm for accelerating the convergence of sequence of functions is demonstrated the relationship of the improved functional epsilon algorithm with the integral pade approximant is also demonstrated moreover we illustrate the similarity between the integral pade approximant and the modified aitken 2 algorithm thus we have shown that the integral pade approximant is a natural generalisation of modified aitken 2 algorithm

a family of pade type approximants for accelerating the convergence of sequences we describe a collection of pade type methods for accelerating the convergence of sequence of functions the construction and connections of pade s methods with other similar methods are given we examine the effectiveness of these new methods namely integral pade approximant modified pade approximant and squared pade approximant together with the well established methods namely functional pade approximant and classical pade approximant for approximating the characteristic value and corresponding characteristic function estimates of characteristic value and characteristic function derived using integral pade approximants are found to be substantially more accurate than other similar methods

green function for schrodinger operator and conditioned feynman kac gauge consider the schrodinger operator h 2 q in a bounded c1 1 domain d with q klocd d q satisfies the condition sup spec 2 q d v x y g x y e y x exp 0 t q x t d t x y d where xt is the conditioned brownian motion starting at x and ending at y with the life time t and v g can be extended to a continuous and positive function on d d specifically we give the comparison theorem there exists a constant c c d q 0 such that 1 c g x y v x y c g x y x y d another result is that v nz x z x d z d exists and represents the poisson kernel for the dirichlet problem corresponding to schrodinger operator h

boundary harnack principle for symmetric stable processes in this paper we study potential theoretic properties of the symmetric stable processes 0 2 establishing the boundary harnack principle for ratios of harmonic functions on any open sets identifying the martin boundary with the euclidean boundary for open sets with a certain interior fatness property and extending earlier results on intrinsic ultracontractivity and the conditional gauge theorem to certain open sets

solutions with moving singularities for a semilinear parabolic equation abstract we consider the cauchy problem for a semilinear heat equation with power nonlinearity it is known that the equation has a singular steady state in some parameter range our concern is a solution with a moving singularity that is obtained by perturbing the singular steady state by formal expansion it turns out that the remainder term must satisfy a certain parabolic equation with inverse square potential from the well posedness of this equation we see that there appears a critical exponent paying attention to this exponent for a prescribed motion of the singular point and suitable initial data we establish the time local existence uniqueness and comparison principle for such singular solutions we also consider solutions with multiple singularities

feynman kac functionals and positive solutions of 1 2δu qu 0 let d be a bounded c 2 domain in d and let q be a bounded borel function in d for x d and z d suppose x t under the law p x z is brownian motion in d starting at x and conditioned to converge to z let be the lifetime of x t we show that if the quantity e x z left exp left int limits 0 t q x s ds right right is finite for one x d and one z d then this quantity remains bounded as x varies over d and z varies over d this may be considered one quantitative expression of the qualitative statement that no matter where brownian motion in d eventually hits d it goes all over d before it gets there we apply this result to show that if the equation 1 2 u qu 0 admits a non negative solution in d which is strictly positive on a subset of d of positive harmonic measure then for any non negative bounded borel function f on d it admits a unique bounded solution u satisfying u f on d and this solution u is non negative

positive solutions with a time independent boundary singularity of semilinear heat equations in bounded lipschitz domains abstract we study time global positive solutions of semilinear heat equations of the form u t u f x u in a bounded lipschitz domain in r n in particular we show the existence of a positive solution with a time independent singularity at a boundary point of which converges to a positive solution with the behavior like the martin kernel at of the corresponding elliptic equation at time infinity a nonlinear term f is conditioned in terms of a certain lipschitz continuity with respect to the second variable and a generalized kato class associated with the martin kernel at and admits not only usual one v x u p log 1 u q but also one with variable exponents

numerical methods for fredholm integral equations on the square abstract in this paper we shall investigate the numerical solution of two dimensional fredholm integral equations by nystrom and collocation methods based on the zeros of jacobi orthogonal polynomials the convergence stability and well conditioning of the method are proved in suitable weighted spaces of functions some numerical examples illustrate the efficiency of the methods

a martin boundary in the plane let e be an open connected subset of euclidean space with a green function and let a be harmonic measure on the martin boundary i of e we will show that except for a a x a null set of x y g i 2 x is an entrance point for brownian motion conditioned to leave e at y r s martin gave examples in dimension 3 or higher for which there exist minimal accessible martin boundary points x 7 y for which this condition fails we will give a similar example in dimension 2

a fast numerical solution method for two dimensional fredholm integral equations of the second kind based on piecewise polynomial interpolation abstract in this paper we consider fast numerical solution methods for two dimensional fredholm integral equation of the second kind f x y a x y u v f u v du dv g x y x y where a x y u v is smooth and g x y is in l 2 2 discretizing the integral equation by certain quadrature rule we get a linear system to deduce fast approximate solution methods for the resulted linear system we study the approximation of the four variable kernel function a x y u v by piecewise polynomial partition the domain 4 into subdomains of the same size and interpolate the kernel function a x y u v in each subdomain fast matrix vector multiplication algorithms and efficient iterative methods are derived numerical results are given to illustrate the efficiency of our methods

conditional transformation of drift formula and potential theory for 1 2 b using conditional brownian motion and the transformation of drift formula of cameron martin girsarov maruyama we give integral conditions on a vector fieldb which imply the harmonic measures and green functions for 1 2 and 1 2 b on a bounded lipschitz domaind are equivalent by equivalent we mean there exist two sided inequalities with constants depending only onb andd this enables one to conclude the potential theory for 1 2 b ond and 1 2 ond are the same

further development of the new algorithms for accelerating the convergence of functional type sequences abstract there are two aims of this paper firstly we shall introduce the determinantal formulae for the new functional type algorithms and secondly we shall demonstrate further development of the new functional type algorithms namely algorithms and algorithms we consider the use of the new algorithms to accelerate the convergence of functional type sequences and their effectiveness for approximating the solution of a given power series is illustrated in process we shall demonstrate the convergence of each of the methods considered the estimates derived using the algorithms and algorithms are found to be very comparative with other similar type of algorithms

richardson extrapolation of iterated discrete galerkin solution for two dimensional fredholm integral equations in this paper we study the numerical solution of two dimensional fredholm integral equation by discrete galerkin and iterated discrete galerkin method we are able to derive an asymptotic error expansion of the iterated discrete galerkin solution this expansion covers arbitrarily high powers of the discretization parameters if the solution of the integral equation is smooth the expansion gives rise to richardson type extrapolation schemes which rapidly improve the original rate of the convergence numerical experiments confirm our theoretical results

generalized 3g theorem and application to relativistic stable process on non smooth open sets let g x y and gd x y be the green functions of rotationally invariant symmetric stable process in rd and in an open set d respectively where 0 2 the inequality gd x y gd y z gd x z c g x y g y z is a very useful tool in studying local schrodinger operators when the above inequality is true with c c d 0 then we say that the 3g theorem holds in d in this paper we establish a generalized version of 3g theorem when d is a bounded fat open set which includes a bounded john domain the 3g we consider is of the form gd x y gd z w gd x w where y may be different from z when y z we recover the usual 3g the 3g form gd x y gd z w gd x w appears in non local schrodinger operator theory using our generalized 3g theorem we give a concrete class of functions belonging to the non local kato class introduced by chen and song on fat open sets as an application we discuss relativistic stable processes relativistic hamiltonian when 1 in fat open sets we identify the martin boundary and the minimal martin boundary with the euclidean boundary for relativistic stable processes in fat open sets furthermore we show that relative fatou type theorem is true for relativistic stable processes in fat open sets the main results of this paper hold for a large class of symmetric markov processes as are illustrated in the last section of this paper we also discuss the generalized 3g theorem for a large class of symmetric stable levy processes

conditioned brownian motion in planar domains we give an upper bound for the green functions of conditioned brownian motion in planar domains a corollary is the conditional gauge theorem in bounded planar domains

local holder continuity of weak solutions to a diffusive shallow medium equation abstract we show the local holder continuity of bounded weak solutions to a doubly nonlinear parabolic equation used in models for diffusion of shallow media

existence for evolutionary problems with linear growth by stability methods abstract we establish that solutions to the cauchy dirichlet problem t u div d f x d u 0 for functionals f r n n 0 of linear growth can be obtained as limits of solutions to flows with p growth in the limit p 1 the result can be interpreted on the one hand as a stability result on the other hand it provides an existence result for general flows with linear growth

a variational approach to parabolic equations under general and p q growth conditions abstract we consider variational solutions to the cauchy dirichlet problem t u div d f x u d u d u f x u d u in t u u 0 on par t where the function f f x u f r n r n r n n 0 is convex with respect to u and coercive in r n n but it not necessarily satisfies a growth condition from above a motivation to consider a class of such energy functions f can be also easily found in the stationary case where a large literature in the calculus of variations is devoted to the minimization of p q growth problems 45 and to double phase problems 23 24 4 5 6 in the parabolic context the notion of variational solution see the references from 8 to 15 is compatible with the lack of the same polynomial growth from below and from above

necessary and sufficient conditions for absolute continuity of elliptic harmonic measure our purpose in this paper is to give a necessary and sufficient condition on the modulus of continuity of the coefficients of an elliptic operator in divergence form in order that the corresponding poisson kernel for the dirichlet problem exist this condition on the coefficients is global continuity together with the additional property that the modulus of continuity along some nontangential direction at each boundary point be bounded uniformly in these points and directions by a function 7 t satisfying the dini type condition

caloric functions in lipschitz domains and the regularity of solutions to phase transition problems in some domain d c rn l that vanishes locally on some distinguished part of od the term refers to the fact that in some sense along that part of od u resembles the fundamental solution of the heat equation in d this is particularly the case when the domain under consideration d is the intersection of some n 1 dimensional cube q with q one side of a lipschitz graph i e q fx f x t that is d q n q and the distinguished part of od is precisely q n q this is the main first area of study of this paper relying on work of fabes garofalo salsa fgs on backward harnack type inequalities for such domains lipschitz regularity in time versus lipschitz regularity in space is not of course the natural homogeneity balance for the study of parabolic equations but it is so for the study of phase transition relations of the form

weighted energy dissipation functionals for doubly nonlinear evolution abstract this paper is concerned with the weighted energy dissipation wed functional approach to doubly nonlinear evolutionary problems this approach consists in minimizing wed functionals defined over entire trajectories we present the features of the wed variational formalism and analyze the related euler lagrange problems moreover we check that minimizers of the wed functionals converge to the corresponding limiting doubly nonlinear evolution finally we present a discussion on the functional convergence of sequences of wed functionals and present some application of the abstract theory to nonlinear pdes

solution of integral equations using generalised inverse function valued pade approximants i abstract neumann series solutions for the linear integral equation x 1 1 0 1 x y y d y converge for small values of we use generalised inverse function valued pade approximants to accelerate the convergence of the neumann series and to estimate the singular value of these estimates are found to compare very favourably with those from the fredholm determinant method and the pade approximant method

the obstacle problem for a fourth order semilinear parabolic equation abstract this paper is concerned with the obstacle problem for the l 2 gradient flow for a functional which is higher order non convex and unbounded from below we prove i the existence and uniqueness of local in time solutions to the obstacle problem and ii a gradient structure of the functional of the solutions via minimizing movements moreover we show the existence of solutions which blow up in a finite time

function valued pade type approximant via e algorithm and its applications in solving integral equations the function valued pade type approximant fpta was defined in the inner product space 8 in this work we choose the coefficients in the neumann power series to make the inner product with both sides a function valued system of equations to yield a scalar system then we express an fpta in the determinant form to avoid the direct computation of the determinants we present the e algorithm for fpta based on the vector valued e algorithm given by brezinski 4 the method of fpta via e algorithm fptavea not only includes all previous methods but overcomes their essential difficulties the numerical experiment for a typical integral equation 1 illustrates that the method of fptavea is simpler and more effective for obtaining the characteristic values and the characteristic functions than all previous methods in addition this method is also applicable to other fredholm integral equations of the second kind without explicit characteristic values and characteristic functions a corresponding example 12 is given and the numerical result is the same as that in 12

vector valued rational interpolants i it is shown explicitly that the samelson inverse may be used to define vector valued thiele type rational interpolants and the equivalence with claessens algorithm is established for the special case of vector valued pade approximants the general form of mccleod s theorem follows as a corollary

global existence and asymptotic behavior of solutions for nonlinear parabolic equations on unbounded domains we study the existence and the asymptotic behavior of positive solutions for the parabolic equation a u tu vup 0 on d 0 where a 0 d is a some unbounded domain in rn n 3 and v belongs to a new parabolic class j of singular potentials generalizing the well known parabolic kato class at infinity p introduced recently by zhang we also show that the choice of this class is essentially optimal

nystrom method for fredholm integral equations of the second kind in two variables on a triangle the paper deals with the approximation of the solution of the following bivariate fredholm integral equationf y email protected dk x y f x w x dx g y email protected d where the domain d is a triangle the proposed procedure by a suitable transformation is essentially the nystrom method based on the zeros of univariate jacobi orthogonal polynomials convergence stability and well conditioning of the method are proved in order to illustrate the efficiency of the proposed method some numerical tests are given

local calderon zygmund estimates for parabolic minimizers abstract we establish local calderon zygmund estimates for parabolic minimizers u to certain variational inequalities the formal corresponding parabolic partial differential equation would be t u div f z u d u u f z u d u div f p 2 f h with p 2 n n 2 however since we do not impose any condition on the existence of the derivative u f the pde above may not have any meaning at all for proving the calderon zygmund estimates we thus remain solely on the level of parabolic minimizers precisely for q p we prove that d u l loc q if u is locally holder continuous and f l loc q and h l loc q for a q depending on q

on a parabolic equation with a singular lower order term abstract we obtain the existence of the weak green s functions of parabolic equations with lower order coefficients in the so called parabolic kato class which is being proposed as a natural generalization of the kato class in the study of elliptic equations as a consequence we are able to prove the existence of solutions of some initial boundary value problems moreover based on a lower and an upper bound of the green s function we prove a harnack inequality for the non negative weak solutions

a variational principle for nonpotential perturbations of gradient flows of nonconvex energies abstract we investigate a variational approach to nonpotential perturbations of gradient flows of nonconvex energies in hilbert spaces we prove existence of solutions to elliptic in time regularizations of gradient flows by combining the minimization of a parameter dependent functional over entire trajectories and a fixed point argument these regularized solutions converge up to subsequences to solutions of the gradient flow as the regularization parameter goes to zero applications of the abstract theory to nonlinear reaction diffusion systems are presented

nonlocal diffusion equations abstract in this paper we present a variational approach to establish the existence and uniqueness of variational solutions to nonlocal evolutionary problems the model we have in mind is the following nonlocal version of the parabolic p laplacian u t div a d u t l p p d u p 2 d u h which has recently attracted many authors we emphasize that our approach also applies in situations where the considered functionals do not allow the derivation of the associated euler lagrange equation

parabolic systems with p q growth a variational approach we consider the evolution problem associated with a convex integrand f mathbb r nn to 0 infty satisfying a non standard p q growth assumption to establish the existence of solutions we introduce the concept of variational solutions in contrast to weak solutions that is mappings u colon omega t to mathbb r n which solve partial tu rm div df du 0 weakly in omega t variational solutions exist under a much weaker assumption on the gap q p here we prove the existence of variational solutions provided the integrand f is strictly convex and frac 2n n 2 p le q p 1 these variational solutions turn out to be unique under certain mild additional assumptions on the data moreover if the gap satisfies the natural stronger assumption 2 le p le q p rm min big 1 frac 4 n big we show that variational solutions are actually weak solutions this means that solutions u admit the necessary higher integrability of the spatial derivative du to satisfy the parabolic system in the weak sense that is we prove that u in l q rm loc big 0 t w 1 q rm loc omega mathbb r n big

on ratios of harmonic functions abstract let u and v be harmonic functions in r n with the same zero set z we show that the ratio f of such functions is always well defined and is real analytic moreover it satisfies the maximum and minimum principles for n 3 we also prove the harnack inequality and the gradient estimate for the ratios of harmonic functions namely sup k f c inf k f sup k f c inf k f for any compact subset k of where the constant c depends on k z only in dimension two the first inequality follows from the boundary harnack principle and the second from the gradient estimate recently obtained by mangoubi it is an open question whether these inequalities remain true in higher dimensions n 4

lifetime of conditioned brownian motion in lipschitz domains if d d d 3 is bounded and has lipschitz boundary then the expected lifetime of any brownian h path process in d is finite

the boundary growth of superharmonic functions and positive solutions of nonlinear elliptic equations we investigate the boundary growth of positive superharmonic functions u on a bounded domain in mathbb r n n 3 satisfying the nonlinear elliptic inequality 0 le delta u le c delta omega x alpha u p quad rm in omega where c 0 0 and p 0 are constants and delta omega x is the distance from x to the boundary of the result is applied to show a harnack inequality for such superharmonic functions also we study the existence of positive solutions with singularity on the boundary of the nonlinear elliptic equation delta u vu f x u quad rm in omega where v and f are borel measurable functions conditioned by the generalized kato class

global gradient estimates for non quadratic vector valued parabolic quasi minimizers abstract we establish local and global higher integrability estimates for the gradient of vector valued parabolic quasi minimizers of integral functionals with polynomial p growth p 2 n n 2 on a parabolic domain t in r n 1 n 2 and under a weak regularity assumption on the parabolic boundary of t

pade type approximation method for two dimensional fredholm integral equations of the second kind two dimensional fredholm integral equation of the second kind 2df ii on a bounded domain d is regarded as the problem with characteristic values a two dimensional function valued pade type approximation 2dfpta is defined its error formulas and convergence theorems are presented to obtain higher order 2dfpta a determinantal expression and its recursive algorithm are given in the end three numerical examples are tested where one on the unit triangle of vertices 0 0 0 1 1 0 and the other two on the square the testing results show that 2dfpta method is more accurate