hadamard inverses square roots and products of almost semidefinite matrices abstract let a aij be an n n symmetric matrix with all positive entries and just one positive eigenvalue bapat proved then that the hadamard inverse of a given by a 1 1 a ij is positive semidefinite we show that if moreover a is invertible then a 1 is positive definite we use this result to obtain a simple proof that with the same hypotheses on a except that all the diagonal entries of a are zero the hadamard square root of a given by a 1 2 a ij 1 2 has just one positive eigenvalue and is invertible finally we show that if a is any positive semidefinite matrix and b is almost positive definite and invertible then a b 1 e t b 1 e a

gradient estimates below the duality exponent we show sharp local a priori estimates and regularity results for possibly degenerate non linear elliptic problems with data not lying in the natural dual space we provide a precise non linear potential theoretic analog of classical potential theory results due to adams duke math j 42 765 778 1975 and adams and lewis studia math 74 169 182 1982 concerning morrey spaces imbedding regularity properties for this we introduce a technique allowing for a non local representation of solutions via riesz potentials in turn yielding optimal local estimates simultaneously in both rearrangement and non rearrangement invariant function spaces in fact we also derive sharp estimates in lorentz spaces covering borderline cases which remained open for some while

fractional hadamard powers of positive semidefinite matrices we consider the class sn of all real positive semidefinite n n matrices and the subclass sn of all a sn with non negative entries for a positive non integer number and some a sn when will the fractional hadamard power a again belong to sn it is known that for a specific this holds for all a sn if and only if n 2 now let a sn be of the form a t v where t sn has rank 1 and v sn has rank p 1 if the hadamard quotient of t and v is hadamard independent in general position and v has sufficently small entries then a complete answer is given depending on n p and special attention is given to the case that p 1

rank of a hadamard product abstract we provide a new proof of a sharp lower bound for the rank of the hadamard product of positive semidefinite matrices our proof uses standard matrix analysis tools and involves the rank of one factor and the kruskal rank of the other it leads to a new understanding of when the hadamard exponential of a positive semidefinite matrix can be singular

marcinkiewicz estimates for degenerate parabolic equations with measure data we consider solutions to degenerate parabolic equations with measurable coefficients having on the right hand side a measure satisfying a suitable density condition we prove integrability results for the gradient in the marcinkiewicz scale

hadamard powers of some positive matrices abstract positivity properties of the hadamard powers of the matrix 1 x i x j for distinct positive real numbers x 1 x n and the matrix cos i j n are studied in particular it is shown that the n n matrix 1 x i x j r is positive semidefinite if and only if r is a nonnegative integer or r n 2 and for every odd integer n 3 the n n matrix cos i j n r is positive semidefinite if and only if r is a nonnegative even integer or r n 3

lorentz estimates for obstacle parabolic problems we prove that the spatial gradient of variational solutions to parabolic obstacle problems of p laplacian type enjoys the same regularity of the data and of the derivatives of the obstacle in the scale of lorentz spaces

universal potential estimates abstract we prove a class of endpoint pointwise estimates for solutions to quasilinear possibly degenerate elliptic equations in terms of linear and nonlinear potentials of wolff type of the source term such estimates allow to bound size and oscillations of solutions and their gradients pointwise and entail in a unified approach virtually all kinds of regularity properties in terms of the given datum and regularity of coefficients in particular local estimates in holder lipschitz morrey and fractional spaces as well as calderon zygmund estimates follow as a corollary in a unified way moreover estimates for fractional derivatives of solutions by mean of suitable linear and nonlinear potentials are also implied the classical wolff potential estimate by kilpelainen mal and trudinger wang as well as recent wolff gradient bounds for solutions to quasilinear equations embed in such a class as endpoint cases

elliptic equations with measurable coefficients in reifenberg domains abstract we prove w 1 p estimates for elliptic equations in divergence form under the assumption that for each point and for each sufficiently small scale there is a coordinate system so that the coefficients have small oscillation in n 1 directions we assume the boundary to be reifenberg flat and the coefficients having small oscillation in the flat direction of the boundary

calderon zygmund theory for nonlinear elliptic problems with irregular obstacles abstract we consider a nonhomogeneous elliptic problem with an irregular obstacle involving a discontinuous nonlinearity over an irregular domain in divergence form of p laplacian type to establish the global calderon zygmund estimate by proving that the gradient of the weak solution is as integrable as both the gradient of the obstacle and the nonhomogeneous term under the bmo smallness of the nonlinearity and sufficient flatness of the boundary in the reifenberg sense

asymptotic stability of a stochastic may mutualism system abstract this paper deals with a mutualism system in random environments in which the cooperation of two species is guaranteed depending on increasing the carrying capacity of each other mathematically we obtain the existence and uniqueness of a stable stationary distribution by means of markov semigroup theory and fokker planck equation in addition we prove that densities of the distributions for the solutions converge in l 1 to an invariant density under appropriate conditions numerical simulations are also presented to illustrate the theoretical results

gradient estimates for the p x laplacean system we prove calderon and zygmund type estimates for a class of elliptic problems whose model is the non homogeneous p x laplacean system div jduj p x 2 du div jf j p x 2 f

lorentz estimates for degenerate and singular evolutionary systems we prove estimates of calderon zygmund type for evolutionary p laplacian systems in the setting of lorentz spaces we suppose the coefficients of the system to satisfy only a vmo condition with respect to the space variable our results hold true mutatis mutandis also for stationary p laplacian systems

a quantitative version of the observation that the hadamard product is a principal submatrix of the kronecker product abstract the hadamard and kronecker products of two n m matrices a b are related by a b p t 1 a b p 2 where p 1 p 2 are partial permutation matrices after establishing several properties of the p matrices this relationship is employed to demonstrate how a simplified theory of the hadamard product can be developed during this process the well known result a b a b aa bb is extended to a b a b 1 2 aa bb ab ba aa bb showing an inherent link between the hadamard product and conventional product of two matrices this leads to a sharper bound on the spectral norm of a b a b 1 2 a 2 b 2 ab 2 1 2 a b and an improvement on the weak majorization of a b 2 a b w 1 2 2 a 2 b 2 ab w 2 a 2 b for a real non singular matrix x and invertible diagonal matrices d e the spectral condition number is shown to be if scaled bounded below as follows dxe 2 x x t 2 x x t 2 1 2 x x t 1 for a 0 we have i a 2 1 2 i a 2 a a i a 2 and a 1 2 a 1 2 2 1 2 i a a 1 a a 1 when a 0 the latter inequality is compared to styan s inequality a a 1 1 2 i a a 1 when a is a correlation matrix and is shown to possess stronger properties of ordering finally the relationship a b p t 1 a b p 2 is applied to determine conditions of singularity of certain orderings of the hadamard products of matrices

gradient estimates for a class of parabolic systems we establish local calderon zygmund type estimates for a class of parabolic problems whose model is the nonhomogeneous degenerate singular parabolic p laplacian system ut div du p 2du div f p 2f proving that f lqloc du lqloc q p we also treat systems with discontinuous coefficients of vanishing mean oscillation vmo type

simultaneous kernels of matrix hadamard powers abstract in previous work belton et al 2016 2 the structure of the simultaneous kernels of hadamard powers of any positive semidefinite matrix was described key ingredients in the proof included a novel stratification of the cone of positive semidefinite matrices and a well known theorem of hershkowitz neumann and schneider which classifies the hermitian positive semidefinite matrices whose entries are 0 or 1 in modulus in this paper we show that each of these results extends to a larger class of matrices which we term 3 pmp principal minor positive

non lipschitz minimizers of smooth uniformly convex functionals we construct non lipschitz minimizers of smooth uniformly convex functionals of type i u f du x dx our method is based on the use of null lagrangians

global gradient estimates for general nonlinear parabolic equations in nonsmooth domains we establish the natural calderon zygmund theory for a nonlinear parabolic equation of p laplacian type in divergence form 0 1 ut diva du x t div f p 2f in t by essentially proving that 0 2 f p lq t du p lq t for every q 1 the equation under consideration is of general type and not necessarily of variation form the involved nonlinearity a a x t is assumed to have a small bmo semi norm with respect to x t variables and the lateral boundary of the domain is assumed to be reifenberg flat as a consequence we are able to not only relax the known regularity requirements on the nonlinearity for such a regularity theory but also extend local results to a global one in a nonsmooth domain whose boundary has a fractal property we also find an optimal regularity estimate in orlicz sobolev spaces for such nonlinear parabolic problems