pointwise gradient estimates for a class of singular quasilinear equations with measure data abstract local and global pointwise gradient estimates are obtained for solutions to the quasilinear elliptic equation with measure data div a x u in a bounded and possibly nonsmooth domain in r n here div a x u is modeled after the p laplacian our results extend earlier known results to the singular case in which 3 n 2 2 n 1 p 2 1 n

pointwise gradient estimates abstract we survey a number of recent results concerning the possibility of proving pointwise gradient estimates via potentials for solutions to quasilinear possibly degenerate elliptic and parabolic equations

non linear elliptic and parabolic equations involving measure data in this paper we prove the existence of solutions for equations of the type div a du f in a bounded open set u 0 on where a is a possibly non linear function satisfying some coerciveness and monotonicity assumptions and f is a bounded measure we also consider the equation div a du g u f in u 0 on with f l1 or f m u u 0 and the parabolic equivalent of the first elliptic equation

a global regularity result for some degenerate elliptic systems abstract we establish a caccioppoli type inequality for a second order degenerate elliptic systems of p laplacian type as a direct consequence exploiting classical campanato s approach and the hole filling technique due to widman we are able to prove a global regularity result on morrey s and l q spaces with q p

nonlinear equations and weighted norm inequalities we study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities in particular we obtain explicit criteria for the solvability of the dirichlet problem u v u w u 0 on u 0 on on a regular domain in rn in the superlinear case q 1 the coefficients v w are arbitrary positive measurable functions or measures on we also consider more general nonlinear differential and integral equations and study the spaces of coefficients and solutions naturally associated with these problems as well as the corresponding capacities our characterizations of the existence of positive solutions take into account the interplay between v w and the corresponding green s kernel they are not only sufficient but also necessary and are established without any a priori regularity assumptions on v and w we also obtain sharp two sided estimates of solutions up to the boundary some of our results are new even if v 1 and is a ball or half space the corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so called 3g inequality by an elementary integration by parts argument this also gives a new unified proof for some classical inequalities including the carleson measure theorem for poisson integrals and trace inequalities for riesz potentials and green potentials

on the sensitivity and relaxability of optimal control problems governed by nonlinear evolution equations with state constraints in this work we examine the relation between the sensitivity well posedness and relaxability for nonlinear distributed parameter systems we introduce the notion of strong calmness which describes the dependence of the value of the problem on the state constraints and we show that it is equivalent to relaxability we also present an alternative control free description of the relaxed problem an example of a nonlinear parabolic optimal control problem of lagrange type is worked out in detail

on the weak continuity of elliptic operators and applications to potential theory in this paper we establish weak continuity results for quasilinear elliptic and subellip tic operators of divergence form acting on corresponding classes of subharmonic functions these results are analogous to our earlier results for fully nonlinear fc hessian operators from the weak continuity we derive various potential theoretic results including capacity estimates potential esti mates and the wiener criterion for regular boundary points our methods make substantial use of harnack inequalities for solutions 1 introduction there are numerous papers on the local behavior of solu tions subsolutions and supersolutions of second order quasilinear elliptic diver gence structure partial differential equations as well as extensions to degenerate elliptic and subelliptic equations in this paper we show that many properties including the quasi continuity of subharmonic functions asymptotic behavior of fundamental solutions removability of singularities wiener criteria for regular boundary points can be derived via harnack inequalities for solutions the two main ingredients in our approach which follow from our treatment of the fully nonlinear hessian operators in 45 46 47 together with that of labutin 30 are a weak convergence result and a pointwise potential estimate for subharmonic functions the weak convergence result is analogous to that for hessian operators 45 46 while the potential estimate extends that found for quasilinear elliptic operators by kilpelainen and maly 27 weak convergence results in the theory of nonlinear partial differential equa tions have been addressed in the survey 13 in this paper we treat nonlinear elliptic divergence structure operators for which harnack type inequalities are available to state our result we need an appropriate notion of subharmonic func tion in the fully nonlinear case 46 we employed the viscosity notion based on differential inequalities for smooth test functions for quasilinear divergence structure equations the traditional definition using the homogeneous equation is more convenient accordingly we shall say that a continuous function w in a domain q in euclidean space ir is harmonic with respect to an operator or c harmonic in short if it is a solution of the homogeneous equation c u 0

mixed inar 1 poisson regression models analyzing heterogeneity and serial dependencies in longitudinal count data this paper presents finite mixture versions of integer valued autoregressive inar poisson regression models for investigating regularity and predictability of purchase behavior over time the approach facilitates the analysis of heterogeneity and serial correlation effects as well as conditional and marginal analyses of the effects of covariates an application to scanner panel data of detergents yields substantive insights into sources of autodependencies in individual category purchases

regularity results for stationary electro rheological fluids we prove regularity results for weak solutions to systems modelling electro rheological fluids in the stationary case as proposed in 27 31 a particular case of the system we consider is

gradient estimates via the wolff potentials for a class of quasilinear elliptic equations abstract in this paper we obtain the pointwise gradient estimates via the nonlinear wolff potentials for weak solutions of a class of non homogeneous quasilinear elliptic equations with measure data

riesz potentials and nonlinear parabolic equations the spatial gradient of solutions to nonlinear degenerate parabolic equations can be pointwise estimated by the caloric riesz potential of the right hand side datum exactly as in the case of the heat equation heat kernels type estimates persist in the nonlinear case

count data distributions some characterizations with applications in this article we characterize all two parameter count distributions satisfying very general conditions that are partially closed under addition we also find those for which the maximum likelihood estimator of the population mean is the sample mean mixed poisson models satisfying these properties are completely determined among these models are the negative binomial poisson inverse gaussian and other known distributions new count distributions can also be constructed using these characterizations three examples of application are given

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

higher order calderon zygmund estimates for the p laplace equation abstract the paper is concerned with higher order calderon zygmund estimates for the p laplace equation div a u div u p 2 u div f 1 p we are able to transfer local interior besov and triebel lizorkin regularity up to first order derivatives from the force term f to the flux a u for p 2 we show that f b q s implies a u b q s for any s 0 1 and all reasonable q 0 in the planar case the result fails for p 2 in case of higher dimensions and systems we have a smallness restriction on s the quasi banach case 0 min q 1 is included since it has important applications in the adaptive finite element analysis as an intermediate step we prove new linear decay estimates for p harmonic functions in the plane for the full range 1 p

gradient estimates via non linear potentials we present pointwise gradient bounds for solutions to p laplacean type non homogeneous equations employing non linear wolff type potentials and then prove similar bounds via suitable caloric potentials for solutions to parabolic equations

on the construction of a parameter orthogonal to the mean it is shown that for many parametric families the condition for a parameter to be orthogonal to the mean takes on a particularly simple form as a result it is often possible to obtain analytically an explicit orthogonal parameter thus providing insight into the many situations involving parametric families in which such a parameter is known to exist orthogonal parameterizations are then given for some other models

weighted gradient estimates for the parabolic p laplacian equations abstract in this paper we obtain the weighted l q estimates of the gradients of weak solutions for the quasilinear parabolic equations of p laplacian type with small bmo coefficients for every e 0 q 1 if u p l w l o c 1 e t and f p l w l o c q t then we prove that u p l w l o c q t our results improve the known results for such equations using a harmonic analysis free technique

bmo estimates for the p laplacian we prove bmo estimates of the inhomogeneous p laplace system given by div jruj p 2 ru divf we show that f 2 bmo impliesjruj p 2 ru2 bmo which is the limiting case of the nonlinear calder on zygmund theory this extends the work of dibenedetto and manfredi dm93 which was restricted to the super quadratic case p 2 to the full case 1 p 1 and even more general growth moreover we prove that a ru inherits the campanato and vmo regularity of f

potential estimates of superquadratic elliptic systems with vmo coefficients in reifenberg domains abstract the purpose of this paper is to study an inhomogeneous dirichlet problem of a superquadratic nonlinear elliptic system with vmo coefficients in reifenberg flat domains we derive pointwise potential estimates for weak solution to the dirichlet problem as a consequence we derive global integrability estimates for solutions with respect to lorentz and morrey norms

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

developments and perspectives in nonlinear potential theory abstract nonlinear potential theory aims at replicating the classical linear potential theory when nonlinear equations are considered in recent years there has been a substantial development of this subject mostly linked to the possibility of proving pointwise estimates for solutions to nonlinear equations via linear and nonlinear potentials here we give a brief account of such developments and outline the connections with different fields

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

estimating the parameters of a convolution by maximum likelihood abstract in the present age of computer sophistication computational difficulty is no longer a justification for seeking alternative and inefficient estimation procedures in place of maximum likelihood estimation particularly when there are only two or three parameters to be estimated convolution densities are an example where this has occurred it is shown that in a large class of such densities the maximum likelihood equations can be reduced by one thus for a two parameter family only a single equation need be solved iteratively the required formulas are derived the problem of basing inferences on the resulting maximum likelihood estimates is briefly discussed

nonlinear calderon zygmund theory in the limiting case we prove a maximal differentiability and regularity result for solutions to nonlinear measure data problems specifically we deal with the limiting case of the classical theory of calderon and zygmund in the setting of nonlinear possibly degenerate equations and we show a complete linearization effect with respect to the differentiability of solutions a prototype of the results obtained here tells for instance that if rm div du p 2 du mu quad mbox in omega subset mathbb r n with mu being a borel measure with locally finite mass on the open subset omega subset mathbb r n and p 2 1 n then du p 2 du in w sigma 1 rm loc omega quad mbox for every sigma in 0 1 the case sigma 1 is obviously forbidden already in the classical linear case of the poisson equation triangle u mu

singular quasilinear and hessian equations and inequalities we solve the existence problem in the renormalized or viscosity sense and obtain global pointwise estimates of solutions for quasilinear and hessian equations with measure coefficients and data including the following model problems pu uq fk u uq u 0 on rn or on a bounded domain rn here p is the p laplacian defined by pu div u u p 2 and fk u is the k hessian i e the sum of the k k principal minors of the hessian matrix d2u k 1 2 n and are general nonnegative measurable functions or measures on

regularity estimates for nonlinear elliptic measure data problems with nonstandard growth abstract we prove a global calderon zygmund type estimate for the gradient of a solution to a nonlinear elliptic problem with nonstandard growth when the right hand side is a bounded radon measure minimal regularity requirements on both the nonlinearity and the boundary of the domain are investigated for such a gradient estimate

nonlinear parabolic problems with lower order terms and related integral estimates abstract we deal with the solutions to nonlinear parabolic equations of the form u t div a x t d u g x t u f x t on t t 0 under standard growth conditions on g and a with f only assumed to be integrable to the power 1 we prove general local decay estimates for level sets of the solutions u and the gradient d u which imply very general estimates in rearrangement function spaces lebesgue orlicz lorentz and non rearrangement ones up to lorentz morrey spaces

global lorentz estimates for nonuniformly nonlinear elliptic equations via fractional maximal operators abstract this article is a contribution to the study of regularity theory for nonlinear elliptic equations the aim of this article is to establish some global estimates for nonuniformly elliptic equations in divergence form as div u p 2 u a x u q 2 u div f p 2 f a x f q 2 f which arises from double phase functional problems in particular the main results provide the regularity estimates for the distributional solutions in terms of maximal and fractional maximal operators this work extends that of colombo and mingione 2016 23 and byun and oh 2017 9 by dealing with the global estimates in lorentz spaces it also extends our recent result tran and nguyen 2020 66 concerning new estimates of divergence elliptic equations using cutoff fractional maximal operators for future research the approach developed in this article will allow global estimates of distributional solutions to nonuniformly nonlinear elliptic equations to be obtained in the framework of other spaces

gradient potential estimates in non linear elliptic obstacle problems with measure data abstract in this paper we derive pointwise estimates for the gradients of solutions to elliptic obstacle problems of p laplace type with measure data moreover we address a borderline problem in the regularity theory and give a sharp criterion for continuity of the gradient of a solution other applications are lorentz space estimates for the gradient and a criterion for c 1 regularity of solutions

characterizing additively closed discrete models by a property of their maximum likelihood estimators with an application to generalized hermite distributions this article reports on two parameter count distributions satisfying very general conditions that are closed under addition so that their maximum likelihood estimator mle of the population mean is the sample mean the most important of these in practice the generalized hermite distribution is analyzed and a necessary and sufficient condition is given to ensure that the mle is the solution of likelihood equations score test to contrast the poisson assumption is studied and two examples of applications are given

riesz potential type estimates for nonlinear elliptic equations abstract for p 2 we study nonlinear elliptic equations of p laplacian type with measure data div a d u u x in when the nonlinearity depends on the variable of the possibly unbounded solution itself we obtain the point wise gradient bound and the gradient continuity in terms of a certain riesz potential of the right hand side measure

nonlinear potential theory of elliptic systems abstract we give a summary of recent results from nonlinear potential theory focusing on linear and nonlinear potential estimates of solutions to non homogeneous equations and systems we start with the cases of quasilinear possibly degenerate equations of p laplacian type and passing through fully nonlinear elliptic equations finally move to systems in this last case we describe recent results implying potential estimates for the p laplacian system finally we describe results bridging nonlinear potential theory and classical partial regularity theory for general elliptic systems the main outcomes are new e regularity criteria involving both excess functionals and nonlinear potentials

modelling hospital length of stay using convolutive mixtures distributions length of hospital stay los is an important indicator of the hospital activity and management of health care the skewness in the distribution of los poses problems in statistical modelling because it fails to adequately follow the usual traditional distribution of positive variables such as the log normal distribution we present in this paper a model using the convolution of two distributions a technique well known in the signal processing community the specificity of that model is that the variable of interest is considered to be the resulting sum of two random variables with different distributions one of the variables features the patient related factors in terms of their need to recover from their admission condition while the other models the hospital management process such as the discharging process two estimation procedures are proposed one is the classical maximum likelihood while the other relates to the expectation maximization algorithm we present some results obtained by applying this model to a set of real data from a group of hospitals in victoria australia copyright 2016 john wiley sons ltd

linear potentials in nonlinear potential theory pointwise gradient bounds via riesz potentials such as those available for the linear poisson equation actually hold for general quasilinear degenerate equations of p laplacean type the regularity theory of such equations completely reduces to that of the classical poisson equation up to the c1 level

properties of the relaxed trajectories of evolution equations and optimal control this paper considers a class of nonconvex control problems in a banach space governed by a semilinear evolution equation and establishes the existence of admissible pairs then the introduction of measure valued controls convexifies the system and under general assumptions it is shown that the set of trajectories of the original system is dense in an appropriate topology in the set of relaxed trajectories some useful topological properties of the set of relaxed trajectories are also determined furthermore some optimization problems are solved and in many cases the values of the original and relaxed problems are shown to be equal finally another relaxation result is proved for a different class of systems described by a general nonlinear evolution equation

gradient estimates via linear and nonlinear potentials we prove new potential and nonlinear potential pointwise gradient estimates for solutions to measure data problems involving possibly degenerate quasilinear operators whose prototype is given by pu in particular no matter the nonlinearity of the equations considered we show that in the case p 2 a pointwise gradient estimate is possible using standard linear riesz potentials the proof is based on the identification of a natural quantity that on one hand respects the natural scaling of the problem and on the other allows to encode the weaker coercivity properties of the operators considered in the case p 2 in the case p 2 we prove a new gradient estimate employing nonlinear potentials of wolff type

riesz potential estimates for a class of double phase problems abstract a double phase problem with a bounded borel measure on the right hand side is studied we prove an optimal pointwise gradient estimate for such a measure data problem via riesz potentials of the measure under log dini continuity assumption on the modulating coefficients as a consequence we find an optimized c 1 regularity criterion

good lambda and muckenhoupt wheeden type bounds in quasilinear measure datum problems with applications weighted good lambda type inequalities and muckenhoupt wheeden type bounds are obtained for gradients of solutions to a class of quasilinear elliptic equations with measure data such results are obtained globally over sufficiently flat domains in mathbb r n in the sense of reifenberg the principal operator here is modeled after the p laplacian where for the first time singular case frac 3n 2 2n 1 p le 2 frac 1 n is considered those bounds lead to useful compactness criteria for solution sets of quasilinear elliptic equations with measure data as an application sharp existence results and sharp bounds on the size of removable singular sets are deduced for a quasilinear riccati type equation having a gradient source term with linear or super linear power growth

pseudo maximum likelihood estimation theory and applications abstract pseudo maximum likelihood estimation easily extends to k parameter models and is of interest in problems in which the likelihood surface is ill behaved in higher dimensions but well behaved in lower dimensions several signal plus noise or convolution models are examined which exhibit such behavior and satisfy the regularity conditions of the asymptotic theory for specific models a numerical comparison of asymptotic variances suggests that a psuedo maximum likelihood estimate of the signal parameter is uniformly more efficient than estimators that have been advanced by previous authors a number of other potential applications are noted

lower semicontinuity of integral functionals with nonconvex integrands by relaxation compactification a new approach to the lower semicontinuity of integral functionals is presented by a topological embedding of the control and state spaces in the hilbert cube and a simultaneous relaxation of the control functions a powerful approach emerges whose main features include i a generalized convexity condition is imposed upon the integrand of which the classical convexity condition is a special case ii in the embedded setting the integrand can be supposed lipschitz continuous in control and state arguments without loss of generality iii convergence in measure of the trajectories metamorphoses into l 1 norm convergence in the embedded setting

good λ type bounds of quasilinear elliptic equations for the singular case abstract in this paper we study the good type bounds for renormalized solutions to nonlinear elliptic problem d i v a x u in u 0 on where r n is a finite radon measure and a is a monotone carathedory vector valued function defined on w 0 1 p the operator a satisfies growth and monotonicity conditions and the p capacity uniform thickness condition is imposed on r n for the singular case 3 n 2 2 n 1 p 2 1 n in fact the same good type estimates were also studied by quoc hung nguyen and nguyen cong phuc for instance in nguyen and phuc 0000 and nguyen 0000 authors method was also confined to the case of 3 n 2 2 n 1 p 2 1 n but under the assumption of is the reifenberg flat domain and the coefficients of a have small bmo bounded mean oscillation semi norms otherwise the same problem was considered in phuc 2014 in the regular case of p 2 1 n in this paper we extend their results taking into account the case 3 n 2 2 n 1 p 2 1 n and without the hypothesis of reifenberg flat domain on and small bmo semi norms of a moreover in rest of this paper we also give the proof of the boundedness property of maximal function on lorentz spaces and also the global gradient estimates of solution