w2 p estimates for the parabolic monge ampere equation when u is a solution to the equation ut det dx2u f with f positive continuous and ft satisfying certain growth conditions we establish estimates in l for ut and show that dx2u satisfies uniform interior estimates in lp for 0 p

boundary regularity on the parabolic monge ampere equation abstract this paper establishes boundary estimates for solutions to the parabolic monge ampere equation u t det d 2 u 1 n f x t

bispectrum estimation a parametric approach higher order spectra contain information about random processes that is not contained in the ordinary power spectrum such as the degree of nonlinearity and deviations from normality estimation of the bispectrum which is a third order spectrum has been applied in various fields to obtain information regarding quadratic phase coupling among harmonic components and non gaussianness of processes existing methods of bispectrum estimation are patterned after the conventional methods of power spectrum estimation which are known to possess certain limitations the paper proposes a parametric approach to bispectrum estimation based on a non gaussian white noise driven autoregressive ar model the ar parameter estimates are obtained by solving the third order recursion equations which may be toeplitz in form but not symmetric it is shown that the method provides bispectral estimates that are far superior to the conventional estimates in terms of bispectral fidelity and that in the case of detecting phase coupling among sinusoids the method provides significantly better resolution

gauss curvature flow the fate of the rolling stones we prove firey s 1974 conjecture that convex surfaces moving by their gauss curvature become spherical as they contract to points

calderon zygmund estimates and non uniformly elliptic operators abstract we consider a class of non uniformly nonlinear elliptic equations whose model is given by div d u p 2 d u a x d u q 2 d u div f p 2 f a x f q 2 f where p q and a x 0 and establish the related nonlinear calderon zygmund theory in particular we provide sharp conditions under which the natural and optimal calderon zygmund type result f p a x f q l loc d u p a x d u q l loc holds for every 1 these problems naturally emerge as euler lagrange equations of some variational integrals introduced and studied by marcellini 41 and zhikov 53 in the framework of homogenisation and lavrentiev phenomenon

a semiadaptive smoothing algorithm in bispectrum estimation fourier transform is used to calculate the third order periodogram which is an unbiased estimation of the bispectrum however the fourier transform has energy leakage at adjacent frequencies and the third order periodogram is not a consistent estimator of the bispectrum because the variance does not converge toward zero when the sample size increases to solve these problems we used the kolmogorov zurbenko fourier transform kzft to calculate the third order periodogram and then adaptively smoothed it using a square window with varying bandwidth at each paired frequencies the bandwidth of the square window is expanded until the squared variation within the window reached a prespecified percentage of the total squared variation the algorithm gave good results on simulated data separating two close frequency components with very different energies in noisy data

the uncertainty principle on considere l existence et la regularite des solutions d equations aux derivees partielles la construction de solutions fondamentales explicites et les valeurs propres d operateurs de schrodinger

exterior problems for more general parabolic monge ampere equation in more general domain abstract in this paper we consider the exterior problem for more general parabolic monge ampere equation u t logdet d 2 u in more general domain r t 1 2 n 1 d r t 1 2 n 1 r n 1 t 1 t 2 d t 1 t 2 using the perron method we obtain the existence and uniqueness of viscosity solution with asymptotic behavior at infinity the results of exterior problems for the parabolic monge ampere equation u t logdet d 2 u 1 are extended

on the operator l f f log f abstract the present note deals with the operator l h 1 r n d r n which takes a given element f of the hardy space to the function f log f in general this function need not be locally integrable nevertheless due to peculiar cancellations of large positive and negative terms in the integral f log f with c 0 r n we are able to give meaning to f log f as a schwartz distribution we find several alternatives for this interpretation of f log f

geometric properties of the sections of solutions to the monge ampere equation in this paper we establish several geometric properties of the cross sections of generalized solutions 4 to the monge ampere equation det d2 when the measure t satisfies a doubling property a main result is a characterization of the doubling measures t in terms of a geometric property of the cross sections of 4 this is used to obtain estimates of the shape and invariance properties of the cross sections that are valid under appropriate normalizations 0 introduction let ri r be a convex function a supporting hyperplane to q at the point xo q xo is an affine function x q xo p x xo such that q x x for all x e r given t 0 a section of 0 at height t is the convex set so xo p t x e r x p ri defined by vq xo p q x xo p x xo vx e rn if e c rn then v e uxee vq x by a classical theorem of aleksandrov the class of sets e such that v e is lebesgue measurable is a o algebra that contains the borel sets and one can define the monge ampere measure associated with q as the borel measure u given by pt v e see ch y the purpose in this paper to analyze in detail several important geometric properties of the sections of the convex function q when its associated monge ampere measure u satisfies a doubling condition the interest in these properties comes from the study of the solutions of the monge ampere equation and its linearizations both elliptic and parabolic and from real harmonic analysis see cl c2 c g1 c g2 and h some of these properties have been used in those references to prove a lemma of besicovitch s type and a calderon zygmund decomposition ir terms of sections which allows us to establish estimates of the solutions to the linearized monge ampere equation also some of these properties imply that r received by the editors june 9 1997 1991 mathematics subject classification primary 35j60 35d10 secondary 26b25

pointwise shape adaptive dct for high quality denoising and deblocking of grayscale and color images the shape adaptive discrete cosine transform sa dct transform can be computed on a support of arbitrary shape but retains a computational complexity comparable to that of the usual separable block dct b dct despite the near optimal decorrelation and energy compaction properties application of the sa dct has been rather limited targeted nearly exclusively to video compression in this paper we present a novel approach to image filtering based on the sa dct we use the sa dct in conjunction with the anisotropic local polynomial approximation intersection of confidence intervals technique which defines the shape of the transform s support in a pointwise adaptive manner the thresholded or attenuated sa dct coefficients are used to reconstruct a local estimate of the signal within the adaptive shape support since supports corresponding to different points are in general overlapping the local estimates are averaged together using adaptive weights that depend on the region s statistics this approach can be used for various image processing tasks in this paper we consider in particular image denoising and image deblocking and deringing from block dct compression a special structural constraint in luminance chrominance space is also proposed to enable an accurate filtering of color images simulation experiments show a state of the art quality of the final estimate both in terms of objective criteria and visual appearance thanks to the adaptive support reconstructed edges are clean and no unpleasant ringing artifacts are introduced by the fitted transform

regularity results for a class of functionals with non standard growth we consider the integral functional under non standard growth assumptions that we call p x type namely we assume that a relevant model case being the functional under sharp assumptions on the continuous function p x 1 we prove regularity of minimizers energies exhibiting this growth appear in several models from mathematical physics

c 1 α regularity of solutions to parabolic monge ampere equations we study interior c 1 alpha regularity of viscosity solutions of the parabolic monge amp ere equation u t b x t big det d 2 u big p with exponent p 0 and with coefficients b which are bounded and measurable we show that when p is less than the critical power 1 over n 2 then solutions become instantly c 1 alpha in the interior also we prove the same result for any power p0 at those points where either the solution separates from the initial data or where the initial data is c 1 beta

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

the calderon zygmund property for quasilinear divergence form equations over reifenberg flat domains abstract the results by palagachev 2009 3 regarding global holder continuity for the weak solutions to quasilinear divergence form elliptic equations are generalized to the case of nonlinear terms with optimal growths with respect to the unknown function and its gradient moreover the principal coefficients are discontinuous with discontinuity measured in terms of small b m o norms and the underlying domain is supposed to have fractal boundary satisfying a condition of reifenberg flatness the results are extended to the case of parabolic operators as well

elliptic equations with bmo coefficients in reifenberg domains the inhomogeneous dirichlet problems concerning divergence form elliptic equations are studied optimal regularity requirements on the coefficients and domains for the w1 p theory 1 p are obtained the principal coefficients are supposed to be in the john nirenberg space with small bmo seminorms the domain is a reifenberg domain these conditions for the w1 p theory not only weaken the requirements on the coefficients but also lead to a more general geometric condition on the domains in fact these domains might have fractal dimensions 2004 wiley periodicals inc

interior w 2 p estimates for solutions of the monge ampere equation in this work we adapt the techniques developed in c1 to prove interior estimates for solutions of perturbations of the monge ampere equation

elliptic equations with singular bmo coefficients in reifenberg domains abstract w 1 p estimate for the solutions of elliptic equations whose coefficient matrix can have large jump along the boundary of subdomains is obtained the principal coefficients are supposed to be in the john nirenberg space with small bmo seminorms and the domain and subdomains are reifenberg flat domains moreover it has been shown that the estimates are uniform with respect to the distance between the subdomains

integral operators on the halfspace in generalized lebesgue spaces lp part ii in this paper we generalize a version of the classical calderon zygmund theorem on principle value integrals in generalized lebesgue spaces lp proved in j reine angew math 563 2003 197 220 to kernels which do not satisfy standard estimates on rd 1 this result will be used in part ii of this paper to prove the classical theorem on halfspace estimates of agmon douglis and nirenberg comm pure appl math 12 1959 623 727 for generalized lebesgue spaces lp

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

properties of the solutions to the monge ampere equation abstract we consider solutions to the equation det d 2 when has a doubling property we prove new geometric characterizations for this doubling property by means of the so called engulfing property and deduce the quantitative behaviour of also a constructive approach to the celebrated c1 estimates proved by l caffarelli is presented settling one of the open questions posed by villani amer math soc 58 2003

l q estimates of gradients for evolutional p laplacian systems abstract we study l q estimates of gradients for evolutional p laplacian systems with discontinuous coefficients and external force given by the divergence of bmo functions the result obtained here may be regarded as a partial generalization of the calderon zygmund theory in l q estimate for the parabolic operator of second order to the evolutional p laplace operator

solvability for schrodinger equations with discontinuous coefficients abstract we consider the l p solvability for divergence and non divergence form schrodinger equations with discontinuous coefficients as an application we give the global morrey regularity for divergence and non divergence form schrodinger operators with vmo coefficients in a bounded domain

an end point orlicz type estimate for nonlinear elliptic equations abstract we study nonlinear elliptic equations of p laplacian type in divergence form to establish a natural calderon zygmund type theory of an orlicz space type where the lebesgue space is the special case with t t q p f p l 1 q 2 r d u p l 1 q r in the previous results the estimates obtained were strictly above the natural exponents and the function such as t t log 1 t was ruled out for the candidate of but with our approach can be selected up to the end point case of the estimates and the functions t t and t t log 1 t can be additionally selected for

a simple generalised crossvalidation method of span selection for periodogram smoothing a consistent estimator for the spectral density of a stationary random process can be obtained by smoothing the periodograms across frequency an important component of smoothing is the choice of the span lee 1997 proposed a span selector that was erroneously claimed to be unbiased for the mean squared error the naive use of mean squared error has some important drawbacks in this context because the variance of the periodogram depends on its mean i e the spectrum we propose a new span selector based on the generalised crossvalidation function derived from the gamma deviance this criterion originally developed for use in fitting generalised additive models utilises the approximate full likelihood of periodograms which asymptotically behave like independently distributed chi squared i e gamma random variables the proposed span selector is very simple and easily implemented simulation results suggest that the proposed span selector generally outperforms those obtained under a mean squared error criterion

on w1 p estimates for elliptic equations in divergence form first we will apply the method to study w 1 p regularity for a nonlinear elliptic operator in divergence form we would like to point out that in the particular case of a linear elliptic equation this method gives an alternative proof to the classical one which uses the general theory of singular integrals the utility of the method that we describe below is that hypotheses a and b are obtained directly by studying the deviation of the coefficients of a from the coefficients of a0 and this is usually not a difficult task a more interesting application of this kind of approximation method is to elliptic homogenization problems for which we obtain results that give w 1 p estimates with a weak hypothesis of regularity in the coefficients for instance we are able to study the following cases

a simple span selector for periodogram smoothing summary one approach to estimating the spectral density of a stationary time series is to smooth the periodogram and one important component of this approach is the choice of the span for smoothing this note proposes a new span selector which is based on unbiased risk estimation the proposed span selector is simple and does not impose strong conditions on the unknown spectrum for example it does not require the unknown spectrum to possess a second derivative which is a typical requirement of most plug in type or spline based methods the finite sample performance of the proposed span selector is illustrated via a small simulation

gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients in this paper we derive global w1 and piecewise c1 estimates for solutions to divergence form elliptic equations with piecewise holder continuous coefficients the novelty of these estimates is that even though they depend on the shape and on the size of the surfaces of discontinuity of the coefficients they are independent of the distance between these surfaces

a localization property of viscosity solutions to the monge ampere equation and their strict convexity the purpose of this note is to show a localization property of convex viscosity solutions to the monge ampere inequality 0 1 2 n are strictly convex

gradient estimates for p laplacian equation in composite reifenberg domains abstract we obtain a local gradient estimate for weak solutions of p laplacian type equation in a non smooth domain composed of a finite number of disjoint subdomains with 0 r reifenberg flat boundaries this kind of domain was called 0 r composite reifenberg domain which allows certain subdomain to exist with highly non smooth boundaries worse than reifenberg domain our results improve the known results for such equations

on jorgens calabi and pogorelov type theorem and isolated singularities of parabolic monge ampere equations abstract in the paper we extend jorgens calabi and pogorelov s theorem on entire solutions of elliptic monge ampere equations to parabolic equations associated with gauss curvature flows our results include gutierrez and huang s previous work as a special case besides we also treat the isolated singularities for parabolic monge ampere equations that was firstly studied by jorgens for elliptic case in two dimensions

on the regularity theory of fully nonlinear parabolic equations ii recently m crandall and p l lions 3 developed a very successful method for proving the existence of solutions of nonlinear second order partial differential equations their method called the theory of viscosity solutions also applies to fully nonlinear equations in which even the second order derivatives can enter in nonlinear fashion solutions produced by the viscosity method are guaranteed to be continuous but not necessarily smooth here we announce smoothness results for viscosity solutions our methods extend those of 1 we obtain krylov safonov i e c estimates 8 c 1 schauder c and w estimates for viscosity solutions of uniformly parabolic equations in general form the results can be viewed as a priori estimates on the classical c solutions our method produces in particular regularity results for a broad new array of nonlinear heat equations including the bellman equation 6

regularity results on the parabolic monge ampere equation with vmo type data abstract this paper establishes interior estimates for l p norms orlicz norms of solutions to the parabolic monge ampere equation u t det d 2 u f x t provided that f x t is positive bounded and satisfies a vmo type condition and some integrability conditions for f t our results improve the corresponding results in gutierrez and huang 2001 8 in some sense

interior and boundary w1 q estimates for quasi linear elliptic equations of schrodinger type abstract we consider nonlinear elliptic equations that are naturally obtained from the elliptic schrodinger equation u v u 0 in the setting of the calculus of variations and obtain l q estimates for the gradient of weak solutions in particular we generalize a result of shen 1995 39 in the nonlinear setting by using a different approach this allows us to consider discontinuous coefficients with a small bmo semi norm and non smooth boundaries which might not be lipschitz continuous

on a prioric 1 α andw 2 p estimates for a parabolic monge ampere equation in the gauss curvature flows this paper establishes holder estimates of du and l p estimates of d 2 u for solutions u to the parabolic monge ampere equation au t det d 2 u 1 n

hessian estimates for fourth order elliptic systems with singular bmo coefficients in composite reifenberg domains abstract we consider fourth order elliptic systems in divergence form with greatly discontinuous coefficients in a nonsmooth domain our domain is composed of a finite number of disjoint subdomains with r reifenberg flat boundaries and the coefficients which are allowed to have great discontinuity across the boundaries of subdomains have small bmo semi norms on each subdomain our main result is deriving the global w 2 p 1 p regularity with the estimates independent of the distance between the subdomains

sharp regularity results on second derivatives of solutions to the monge ampere equation with vmo type data we establish interior estimates for lp norms orlicz norms and mean oscillation of second derivatives of solutions to the monge ampere equation det d2u f x with zero boundary value where f x is strictly positive bounded and satisfies a vmo type condition these estimates develop the regularity theory of the monge ampere equation in vmo type spaces our orlicz estimates also sharpen caffarelli s celebrated w2 p estimates 2008 wiley periodicals inc

global estimates for quasilinear elliptic equations on reifenberg flat domains new global regularity estimates are obtained for solutions to a class of quasilinear elliptic boundary value problems the coefficients are assumed to have small bmo seminorms and the boundary of the domain is sufficiently flat in the sense of reifenberg the main regularity estimates obtained are in weighted lorentz spaces other regularity results in lorentz morrey morrey and holder spaces are shown to follow from the main estimates

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

global gradient estimates for nonlinear equations of p laplace type from composite materials abstract we study global gradient estimates for the weak solution to divergence form nonlinear elliptic equations with p growth nonlinearities from composite materials we assume that the domain is bounded and is composed of disjoint reifenberg flat domains and the nonlinearities have small bmo seminorms in each subdomain under these assumptions based on our new geometric observation for disjoint reifenberg domains and gradient estimates for nonlinear equations with measurable p growth nonlinearities we establish global w 1 q estimates for p q