on the stochastic porous medium equation abstract in this paper we discuss an initial boundary value problem and a cauchy problem for the stochastic porous medium equation our basic estimates are based on the known results due to math sbornik 67 1965 609 642 arch rational mech anal 25 1967 64 80 by the procedure developed in trans amer math soc 354 2001 1117 1135 we obtain solutions over the given probability space rather than martingale solutions we will also establish the existence of invariant measures when the space domain is bounded

small mass expansion of functional determinants on the generalized cone in this paper we compute the small mass expansion for the functional determinant of a scalar laplacian defined on the bounded generalized cone in the framework of zeta function regularization we obtain an expression for the functional determinant valid in any dimension for both dirichlet and robin boundary conditions in terms of the spectral zeta function of the base manifold moreover as a particular case we specify the base to be a d dimensional sphere and present explicit results for d 2 3 4 5

the numerical range of a tridiagonal operator abstract let r be a real number and a a tridiagonal operator defined by a e j e j 1 r j e j 1 j 1 2 where e 1 e 2 is the standard orthonormal basis for l 2 n such tridiagonal operators arise in rogers ramanujan identities in this paper we study the numerical ranges of these tridiagonal operators and finite dimensional tridiagonal matrices in particular when r 1 the numerical range of the finite dimensional tridiagonal matrix is the convex hull of two explicit ellipses applying the result we obtain that the numerical range of the tridiagonal operator is the square z c 1 r z 1 1 z 1 1 i 1 i 1 i 1 i

asymptotic couplings by reflection and applications for nonlinear monotone spdes abstract asymptotic couplings by reflection are constructed for a class of nonlinear monotone spdes stochastic partial differential equations as applications gradient holder estimates as well as exponential convergence are derived for the associated markov semigroup the main results are illustrated by stochastic generalized porous media equations stochastic p laplacian equations and stochastic generalized fast diffusion equations we emphasize that the gradient estimate is studied for the first time for these equations

heat kernel coefficients of the laplace operator on the d dimensional ball we present a very quick and powerful method for the calculation of heat kernel coefficients it makes use of rather common ideas as integral representations of the spectral sum mellin transforms non trivial commutation of series and integrals and skillful analytic continuation of zeta functions on the complex plane we apply our method to the case of the heat kernel expansion of the laplace operator on a d dimensional ball with either dirichlet neumann or in general robin boundary conditions the final formulas are quite simple using this case as an example we illustrate in detail our scheme which serves for the calculation of an in principle arbitrary number of heat kernel coefficients in any situation when the basis functions are known we provide a complete list of new results for the coefficients b3 b10 corresponding to the d dimensional ball with all the mentioned boundary conditions and d 3 4 5

the spectral zeta function for laplace operators on warped product manifolds of the type i f n in this work we study the spectral zeta function associated with the laplace operator acting on scalar functions defined on a warped product of manifolds of the type i f n where i is an interval of the real line and n is a compact d dimensional riemannian manifold either with or without boundary starting from an integral representation of the spectral zeta function we find its analytic continuation by exploiting the wkb asymptotic expansion of the eigenfunctions of the laplace operator on m for which a detailed analysis is presented we apply the obtained results to the explicit computation of the zeta regularized functional determinant and the coefficients of the heat kernel asymptotic expansion

weak solutions to the stochastic porous media equation via kolmogorov equations the degenerate case a stochastic version of the porous medium equation with coloured noise is studied the corresponding kolmogorov equation is solved in the space l 2 h where is an infinitesimally excessive measure then a weak solution is constructed

on geometric properties of the numerical range abstract we investigate the shape of the numerical range a criterion for the numerical range of a matrix to be an elliptical disk is given the result is applied to show that there exist neither 3 by 3 nor 4 by 4 nilpotent matrices whose numerical range is an elliptical noncircular disk sufficient conditions for n by n tridiagonal matrices to have elliptical numerical range are obtained the boundary of the numerical range near a sharp point is examined finally the numerical range of a reducible matrix is compressed and its geometric properties are discussed

on a random scaled porous media equation abstract it is shown that a random scaled porous media equation arising from a stochastic porous media equation with linear multiplicative noise through a random transformation is well posed in l in the fast diffusion case we show existence in l p

strong solutions for a stochastic model of two dimensional second grade fluids driven by levy noise abstract in this paper we consider stochastic second grade fluids driven by levy noise on a bounded domain of r 2 global existence and uniqueness of strong probabilistic solutions are established which improves the existing corresponding results for martingale solutions in literature

spectral geometry of singular riemannian spaces on considere la trace du noyau de la chaleur et l application de la methode de l equation de la chaleur au theoreme d indice pour la caracteristique d euler et des complexes signatures

stochastic generalized porous media and fast diffusion equations abstract we present a generalization of krylov rozovskii s result on the existence and uniqueness of solutions to monotone stochastic differential equations as an application the stochastic generalized porous media and fast diffusion equations are studied for finite reference measures where the drift term is given by a negative definite operator acting on a time dependent function which belongs to a large class of functions comparable with the so called n functions in the theory of orlicz spaces

on the numerical ranges of some tridiagonal matrices abstract for certain tridiagonal matrices of small size we give a complete description of flat portions on the boundary of their numerical range we also discuss the conditions for these numerical ranges to be elliptical

a trotter type result for the stochastic porous media equations abstract this paper proves the continuous dependence with respect to diffusivity of the solutions to stochastic porous media equations with noncoercive monotone diffusivity functions and multiplicative noise

an inverse indefinite numerical range problem abstract consider c n with a krein space structure with respect to the indefinite inner product x y x j y x y c n where j is an indefinite self adjoint involution the krein space numerical range w j t of a complex matrix t is the set of all the values attained by the quadratic form t u u where u c n satisfies u u 1 the main aim of this paper is the investigation of the following inverse problem given a complex matrix t and a point z in w j t determine a unit vector that generates z the number of linearly independent generating vectors of z is determined an algorithm for solving the inverse problem is developed implemented and tested

conical casimir pistons with hybrid boundary conditions in this paper we compute the casimir energy and force for massless scalar fields endowed with hybrid boundary conditions in the setting of the bounded generalized cone by using spectral zeta function regularization methods we obtain explicit expressions for the casimir energy and force in arbitrary dimensions in terms of the zeta function defined on the piston our general formulas are subsequently specialized to the case in which the piston is modeled by a d dimensional sphere in this particular situation explicit results are given for d 2 3 4 5

the numerical range of banded biperiodic toeplitz operators abstract the numerical range of an operator is a well studied concept with many applications in several areas of mathematics in this paper the numerical range of banded biperiodic toeplitz operators is investigated performing a reduction to the 2 2 case namely the parametric equations of the boundary generating curves are deduced and two algorithms for the numerical generation of the numerical range are presented

functional determinants on spheres and sectors the scalar functional determinants on sectors of the two dimensional disc and spherical cap are determined for arbitrary angles rational factors of the whole sphere and hemisphere expressions are also given in low dimensions for both the ordinary and the conformal laplacian

harnack inequality and strong feller property for stochastic fast diffusion equations abstract as a continuation to f y wang harnack inequality and applications for stochastic generalized porous media equations ann probab 35 2007 1333 1350 where the harnack inequality and the strong feller property are studied for a class of stochastic generalized porous media equations this paper presents analogous results for stochastic fast diffusion equations since the fast diffusion equation possesses weaker dissipativity than the porous medium one does some technical difficulties appear in the study as a compensation to the weaker dissipativity condition a sobolev nash inequality is assumed for the underlying self adjoint operator in applications some concrete examples are constructed to illustrate the main results

heat kernel coefficients for laplace operators on the spherical suspension in this paper we compute the coefficients of the heat kernel asymptotic expansion for laplace operators acting on scalar functions defined on the so called spherical suspension or riemann cap subjected to dirichlet boundary conditions by utilizing a contour integral representation of the spectral zeta function for the laplacian on the spherical suspension we find its analytic continuation in the complex plane and its associated meromorphic structure thanks to the well known relation between the zeta function and the heat kernel obtainable via mellin transform we compute the coefficients of the asymptotic expansion in arbitrary dimensions the particular case of a d dimensional sphere as the base manifold is studied as well and the first few heat kernel coefficients are given explicitly

well posedness of nonlinear diffusion equations with nonlinear conservative noise we prove the pathwise well posedness of stochastic porous media and fast diffusion equations driven by nonlinear conservative noise as a consequence the generation of a random dynamical system is obtained this extends results of the second author and souganidis who considered analogous spatially homogeneous and first order equations and earlier works of lions perthame and souganidis

heat kernels and functional determinants on the generalized cone we consider zeta functions and heat kernel expansions on the bounded generalized cone in arbitrary dimensions using an improved calculational technique the specific case of a global monopole is analysed in detail and some restrictions thereby placed on thea 5 2 coefficient the computation of functional determinants is also addressed general formulas are given and known results are incidentally and rapidly reproduced

local and global well posedness of spde with generalized coercivity conditions abstract in this paper we establish the local and global existence and uniqueness of solutions for general nonlinear evolution equations with coefficients satisfying some local monotonicity and generalized coercivity conditions an analogous result is obtained for stochastic evolution equations in hilbert space with additive noise as applications the main results are applied to obtain simpler proofs in known cases as the stochastic 3d navier stokes equation the tamed 3d navier stokes equation and the cahn hilliard equation but also to get new results for stochastic surface growth pde and stochastic power law fluids

two point one dimensional delta delta prime interactions non abelian addition law and decoupling limit in this contribution to the study of one dimensional point potentials we prove that if we take the limit on a potential of the type we obtain a new point potential of the type when u0 and u1 are related to v0 v1 w0 and w1 by a law with the structure of a group this is the borel subgroup of we also obtain the non abelian addition law from the scattering data the spectra of the hamiltonian in the decoupling cases emerging in the study are also described in full it is shown that for the values of the couplings the singular kurasov matrices become equivalent to dirichlet at one side of the point interaction and robin boundary conditions at the other side

zeta function determinant of the laplace operator on the d dimensional ball we present a direct approach for the calculation of functional determinants of the laplace operator on balls dirichlet and robin boundary conditions are considered using this approach formulas for any value of the dimension d of the ball can be obtained quite easily explicit results are presented here for dimensionsd 2 3 4 5 and 6

non monotone stochastic generalized porous media equations abstract by using the nash inequality and a monotonicity approximation argument existence and uniqueness of strong solutions are proved for a class of non monotone stochastic generalized porous media equations moreover we prove for a large class of stochastic pde that the solutions stay in the smaller l 2 space provided the initial value does so that some recent results in the literature are considerably strengthened

regularity in sobolev spaces for the fast diffusion and the porous medium equation abstract the degenerate parabolic equation u t u m 1 u m 0 is considered in a cylinder 0 t under homogeneous dirichlet boundary values the regularity of weak solutions for the fast diffusion equation m 1 and the porous medium equation m 1 are investigated regularity of u and u m 1 u in weighted sobolev spaces and in fractional order nikolskii spaces are proved

zeta determinant for laplace operators on riemann caps the goal of this paper is to compute the zeta function determinant for the massive laplacian on riemann caps or spherical suspensions these manifolds are defined as compact and boundaryless d dimensional manifolds deformed by a singular riemannian structure the deformed spheres considered previously in the literature belong to this class after presenting the geometry and discussing the spectrum of the laplacian we illustrate a method to compute its zeta regularized determinant the special case of the deformed sphere is recovered as a limit of our general formulas

on the stochastic p laplace equation the p laplace equation with random perturbation is studied for the singular case 1 p 2 in this paper some properties of the invariant measure and transition semigroups are obtained the main tool is the dimension free harnack inequality which is established by using the coupling argument as consequences some ergodicity compactness and contractive properties are derived for the associated transition semigroups the main results are applied to stochastic reaction diffusion equations and the stochastic p laplace equation in hilbert space