on lp x norms the relation between a banach function space x and its subspaces formed by i those functions with absolutely continuous norm ii those with continuous norm and iii the closure of the set of

introduction to matrix analytic methods in stochastic modeling preface part i quasi birth and death processes 1 examples part ii the method of phases 2 ph distributions 3 markovian point processes part iii the matrix geometric distribution 4 birth and death processes 5 processes under a taboo 6 homogeneous qbds 7 stability condition part iv algorithms 8 algorithms for the rate matrix 9 spectral analysis 10 finite qbds 11 first passage times part v beyond simple qbds 12 nonhomogeneous qbds 13 processes skip free in one direction 14 tree processes 15 product form networks 16 nondenumerable states bibliography index

double phase problems with variable growth abstract we consider a class of double phase variational integrals driven by nonhomogeneous potentials we study the associated euler equation and we highlight the existence of two different rayleigh quotients one of them is in relationship with the existence of an infinite interval of eigenvalues while the second one is associated with the nonexistence of eigenvalues the notion of eigenvalue is understood in the sense of pairs of nonlinear operators as introduced by fucik necas soucek and soucek the analysis developed in this paper extends the abstract framework corresponding to some standard cases associated to the p x laplace operator the generalized mean curvature operator or the capillarity differential operator with variable exponent the results contained in this paper complement the pioneering contributions of marcellini mingione et al in the field of variational integrals with unbalanced growth

on the spaces lp x ω and wm p x ω in this paper we present some basic results on the generalized lebesgue spaces l p x omega and generalized lebesgue sobolev spaces w m w p x omega these results provide the necessary framework for the study of variational problems and elliptic equations with non standard p x growth conditions c 2001 academic press

lifespan of solutions to semilinear damping wave equations in de sitter spacetime abstract this paper studies blowup phenomena for two kind of initial value problem of semilinear damping wave equations in de sitter spacetime the first problem comes from the latter problem of conjecture of strauss in de sitter spacetime and the second problem arises from the latter problem of conjecture of glassey in de sitter spacetime we give blowup and lifespan of solutions on those two problems comparing with non damping case established in yan 2018 one can see that the damping term with a positive bounded time dependent coefficient does not effect blowup of solutions meanwhile we establish a new ode blowup result concerning damping term

on a discrete risk model with two sided jumps in this paper we consider a discrete renewal risk model with phase type interarrival times and two sided jumps in this model downward jumps represent claim loss while upward jumps are also allowed to represent random gains assume that the downward jumps have an arbitrary probability function and the upward jumps have a rational probability generating function we study the gerber shiu discounted penalty function the generating function the recursive formula as well as an explicit expression for the discounted penalty function are obtained

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

boundary harnack inequality for markov processes with jumps we prove a boundary harnack inequality for jump type markov processes on metric measure state spaces under comparability estimates of the jump kernel and urysohn type property of the domain of the generator of the process the result holds for positive harmonic functions in arbitrary open sets it applies e g to many subordinate brownian motions levy processes with and without continuous part stable like and censored stable processes jump processes on fractals and rather general schrodinger drift and jump perturbations of such processes

existence and concentration of semiclassical ground state solutions for the generalized chern simons schrodinger system in h1 r2 abstract this paper is concerned with the following singularly perturbed problem in h 1 r 2 e 2 u v x u a 0 u x u j 1 2 a j 2 u x u f u e 1 a 2 u x 2 a 1 u x 1 2 u 2 1 a 1 u x 2 a 2 u x 0 e a 0 u 1 a 2 u 2 2 a 1 u 2 where e is a small parameter v c r 2 r and f c r r by using some new variational and analytic techniques joined with the manifold of pohozaev nehari type we prove that there exists a constant e 0 0 determined by v and f such that for e 0 e 0 the above problem admits a semiclassical ground state solution v e with exponential decay at infinity we also establish a new concentration behaviour of v e as e 0 in particular our results are available to the nonlinearity f u u s 2 u for s 4 6 which extend the existing results concerning the case f u u s 2 u for s 6

regularity for double phase variational problems we prove sharp regularity theorems for minimisers of a class of variational integrals whose integrand switches between two different types of degenerate elliptic phases according to the zero set of a modulating coefficient a cdot the model case is given by the functional w mapsto int dw p a x dw q rm d x where q p and a cdot geqq 0

unbalanced p 2 fractional problems with critical growth abstract we study the existence multiplicity and regularity results of non negative solutions of following doubly nonlocal problem p s 1 u p s 2 u a x u q 2 u u y r x y d y u r 2 u in u 0 in r n where r n is a bounded domain with c 2 boundary 0 s 2 s 1 1 n 2 s 1 1 q p 2 1 r 2 with 2 2 n n 2 s 1 0 and a l d d q for some q d 2 s 1 2 n n 2 s 1 is a sign changing function we prove that each nonnegative weak solution of p is bounded furthermore we obtain some existence and multiplicity results using nehari manifold method

growth conditions and regularity for weak solutions to nonlinear elliptic pdes abstract we describe some aspects of the process approach to interior regularity of weak solutions to a class of nonlinear elliptic equations in divergence form as well as of minimizers of integrals of the calculus of variations in particular with respect to the growth conditions we emphasize some differences between solving equations and minimizing energy integrals and also between x dependence or x independence

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

potential theory of truncated stable processes for any alpha in 0 2 a truncated symmetric stable process is a symmetric levy process in mathbb r d with a levy density given by c x d alpha 1 x 1 for some constant c in this paper we study the potential theory of truncated symmetric stable processes in detail we prove a harnack inequality for nonnegative harmonic functions of these processes we also establish a boundary harnack principle for nonnegative functions which are harmonic with respect to these processes in bounded convex domains we give an example of a non convex domain for which the boundary harnack principle fails

boundary harnack principle for δ δ 2 for d 1 and 2 0 2 consider the family of pseudo differential operators f b 2 b 2 0 1 g on r d that evolves continuously fromto 2 in this paper we establish a uniform boundary harnack principle bhp with explicit boundary decay rate for nonnegative functions which are harmonic with respect to b 2 or equivalently the sum of a brownian motion and an independent symmetric stable process with constant multipleb 1 inc 1 1 open sets here a uniform bhp means that the comparing constant in the bhp is independent of b 2 0 1 along the way a uniform carleson type estimate is established for nonnegative functions which are harmonic with respect to b 2 in lipschitz open sets our method employs a combination of probabilistic and analytic techniques

a fluid limit for an overloaded x model via a stochastic averaging principle we prove a many server heavy traffic fluid limit for an overloaded markovian queueing system having two customer classes and two service pools known in the call center literature as the x model the system uses the fixed queue ratio with thresholds fqr t control which we proposed in a recent paper as a way for one service system to help another in face of an unexpected overload under fqr t customers are served by their own service pool until a threshold is exceeded then one way sharing is activated with customers from one class allowed to be served in both pools after the control is activated it aims to keep the two queues at a prespecified fixed ratio for large systems that fixed ratio is achieved approximately for the fluid limit or fwlln functional weak law of large numbers we consider a sequence of properly scaled x models in overload operating under fqr t our proof of the fwlln follows the compactness approach i e we show that the sequence of scaled processes is tight and then show that all converging subsequences have the specified limit the characterization step is complicated because the queue difference processes which determine the customer server assignments need to be considered without spatial scaling asymptotically these queue difference processes operate on a faster time scale than the fluid scaled processes in the limit because of a separation of time scales the driving processes converge to a time dependent steady state or local average of a time varying fast time scale process ftsp this averaging principle allows us to replace the driving processes with the long run average behavior of the ftsp

random matrix ensembles with column row constraints ii we analyze statistical properties of a complex system subjected to conditions which manifests through specific constraints on the column row sum of the matrix elements of its hermitian operators the presence of additional constraints besides real symmetric nature leads to new correlations among their eigenfunctions hinders a complete delocalization of dynamics and affects the eigenvalues too the statistical analysis of the latter indicates the presence of a new universality class analogous to that of a special type of brownian ensemble appearing between poisson and gaussian orthogonal ensemble

variable exponent linear growth functionals in image restoration we study a functional with variable exponent 1 leq p x leq 2 which provides a model for image denoising enhancement and restoration the diffusion resulting from the proposed model is a combination of total variation tv based regularization and gaussian smoothing the existence uniqueness and long time behavior of the proposed model are established experimental results illustrate the effectiveness of the model in image restoration

constant sign solutions for double phase problems with superlinear nonlinearity abstract we study parametric double phase problems involving superlinear nonlinearities with a growth that need not necessarily be polynomial based on truncation and comparison methods the existence of two constant sign solutions is shown provided the parameter is larger than the first eigenvalue of the p laplacian as a result of independent interest we prove a priori estimates for solutions for a general class of double phase problems with convection term

queueing simulation in heavy traffic the heavy traffic behavior of a number of standard queueing simulation procedures like sample averaging and regenerative simulation is studied in particular limit theorems based upon brownian approximations are given in a double limit with both the sample size t tending to infinity and the traffic intensity i tending to one the results demonstrate that the growth rate t ti 1 i 2 plays a critical role and it follows both from the present theoretical results and empirical illustrations in a companion paper that negative bias presents a serious problem furthermore the heavy traffic behavior of the limiting variances in the standard large sample theory is found

the boundary behavior of heat kernels of dirichlet laplacians we prove the following complete and qualitatively sharp description of heat kernels g of dirichlet laplacians on bounded c1 1 domains d there exist positive constants c1 c2 and t 0 depending on d such that for x dist x d x y t 1c1tn 2e c2 x y 2 t g x t y 0 x y t 11c1tn 2e x y 2 c2t for all x y d and 0 t are known bounds when d is unbounded are also given

general solution of the poisson equation for quasi birth and death processes we consider the poisson equation i p u g where p is the transition matrix of a quasi birth and death process with infinitely many levels g is a given infinite dimensional vector and u is the unknown our main result is to provide the general solution of this equation to this purpose we use the block tridiagonal and block toeplitz structure of the matrix p to obtain a set of matrix difference equations which are solved by constructing suitable resolvent triples

conditional gauge theorem for non local feynman kac transforms feynman kac transforms driven by discontinuous additive functionals are studied in this paper for a large class of markov processes general gauge and conditional gauge theorems are established for such transforms furthermore the l 2 infinitesimal generator of the schrodinger semigroup given by a non local feynman kac transform is determined in terms of its associated bilinear form

two sided eigenvalue estimates for subordinate processes in domains let x xt t 0 be a symmetric markov process in a state space e and d an open set of e denote by x d the subprocess of x killed upon leaving d let s st t 0 be a subordinator with laplace exponent that is independent of x the processes x xst t 0 and x d x dt t 0 are called the subordinate processes of x and x d respectively under some mild conditions we show that if n n 1 and n n 1 denote the eigenvalues of the generators of the subprocess of x killed upon leaving d and of the process x d respectively then

potential theory of subordinate killed brownian motion in a domain subordination of a killed brownian motion in a bounded domain d d via an 2 stable subordinator gives a process z t whose infinitesimal generator is d 2 the fractional power of the negative dirichlet laplacian in this paper we study the properties of the process z t in a lipschitz domain d by comparing the process with the rotationally invariant stable process killed upon exiting d we show that these processes have comparable killing measures prove the intrinsic ultracontractivity of the generator of z t prove the intrinsic ultracontractivity of the semigroup of z t and in the case when d is a bounded c 1 1 domain obtain bounds on the green function and the jumping kernel of z t

sharp regularity for functionals with p q growth we prove regularity theorems for minimizers of integral functionals of the calculus of variations f x du x dx with non standard growth conditions of p q type z p f x z l z q 1 p q in particular we find that a sufficient condition for minimizers to be regular is qp n n where the function f x z is holder continuous with respect to the x variable and x rn this condition is also sharp we include results in the setting of orlicz spaces moreover we treat certain relaxed functionals too finally we address a problem posed by marcellini in 43 showing a minimizer with an isolated singularity

harnack inequalities for double phase functionals we prove a harnack inequality for minimisers of a class of non autonomous functionals with non standard growth conditions they are characterised by the fact that their energy density switches between two types of different degenerate phases

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

regularity for minimizers of double phase functionals with mild transition and regular coefficients abstract we prove sharp regularity results for minimizers of the functional p w b x w d w p a x d w p log e d w d x with w w 1 1 p 1 a l a 0 and 0 b l p is a double phase functional with mild transition between d u p and d u p log e d u first under suitable conditions on the moduli of continuity of a and b we prove that local minimizers are of class c 0 for every 0 1 then that they are of class c 1 for some 0 provided the functions a and b are holder continuous

positive solutions for singular double phase problems abstract we study the existence of positive solutions for a class of double phase dirichlet equations which have the combined effects of a singular term and of a parametric superlinear term the differential operator of the equation is the sum of a p laplacian and of a weighted q laplacian q p with discontinuous weight using the nehari method we show that for all small values of the parameter 0 the equation has at least two positive solutions

eigenvalue problems with unbalanced growth nonlinear patterns and standing wave solutions abstract we consider two classes of nonlinear eigenvalue problems with double phase energy and lack of compactness we establish existence and non existence results and related properties of solutions our analysis combines variational methods with the generalized pucci serrin maximum principle

two sided green function estimates for killed subordinate brownian motions a subordinate brownian motion is a levy process that can be obtained by replacing the time of the brownian motion by an independent subordinator the infinitesimal generator of a subordinate brownian motion is where is the laplace exponent of the subordinator in this paper we consider a large class of subordinate brownian motions without diffusion component and with comparable to a regularly varying function at infinity this class of processes includes symmetric stable processes relativistic stable processes sums of independent symmetric stable processes sums of independent relativistic stable processes and much more we give sharp two sided estimates on the green functions of these subordinate brownian motions in any bounded fat open set d w hend is a bounded c 1 1 open set we establish an explicit form of the estimates in terms of the distance to the boundary as a consequence of such sharp green function estimates we obtain a boundary harnack principle in c 1 1 open sets with explicit rate of decay

regularity and existence of solutions of elliptic equations with p q growth conditions for some positive constants m a4 and for exponents q 2 p 2 2 under 1 2 1 3 and some other assumptions by assuming also that the quotient q p is sufficiently close to one in dependence on n precisely if q p c n n 2 then we prove that every weak solution to 1 1 of class 4 4 0 is locally lipschitz continuous in q moreover there are positive constants 8 c and 0 2 1 such that

the time dependent expected reward and deviation matrix of a finite qbd process abstract deriving the time dependent expected reward function associated with a continuous time markov chain involves the computation of its transient deviation matrix in this paper we focus on the special case of a finite quasi birth and death qbd process motivated by the desire to compute the expected revenue lost in a map ph 1 c queue we use two different approaches in this context the first is based on the solution of a finite system of matrix difference equations it provides an expression for the blocks of the expected reward vector the deviation matrix and the mean first passage time matrix the second approach based on some results in the perturbation theory of markov chains leads to a recursive method to compute the full deviation matrix of a finite qbd process we compare the two approaches using some numerical examples

higher integrability for constrained minimizers of integral functionals with p q growth in low dimension abstract we prove higher summability for the gradient of minimizers of strongly convex integral functionals of the calculus of variations f x d u x d x u r n s n 1 with growth conditions of p q type p f x c q 1 p q in low dimension our procedure is set in the framework of fractional sobolev spaces and renders the desired regularity as the result of an approximation technique relying on estimates obtained through a careful use of difference quotients

the global behavior of heat kernels in exterior domains abstract we establish the global boundary behavior of dirichlet heat kernels on some unbounded domains these combined with the interior estimate in the recent paper by grigoryan and saloff coste comm pure appl math 55 1 2002 93 provide a complete and qualitatively sharp description of heat kernels g of dirichlet laplacians on some unbounded c1 1 domains d m here m is certain complete noncompact manifold with nonnegative ricci curvature for example there exist positive constants c1 c2 depending on the unbounded domain d m such that for x dist x d x t 1 1 y t 1 1 c 1 e c 2 d x y 2 t b x t g x t y 0 x t 1 1 y t 1 1 e d x y 2 c 2 t c 1 b x t for all x y d and t 0 existence of these global bounds had been an unresolved problem even in the euclidean case

the intercept term of the asymptotic variance curve for some queueing output processes we consider the output processes of some elementary queueing models such as the m m 1 k queue and the m g 1 queue an important performance measure for these counting processes is their variance curve v t which gives the variance of the number of customers in the time interval 0 t recent work has revealed some non trivial properties dealing with the asymptotic rate at which the variance curve grows in this paper we add to these results by finding explicit expressions for the intercept term of the linear asymptote for m m 1 k queues our results are based on the deviation matrix of the generator it turns out that by viewing output processes as markovian point processes and considering the deviation matrix one can obtain explicit expressions for the intercept term together with some further insight regarding the bravo balancing reduces asymptotic variance of outputs effect for m g 1 queues our results are based on a classic transform of d j daley in this case we represent the intercept term of the variance curve in terms of the first three moments of the service time distribution in addition we shed light on a conjecture of daley dealing with characterization of stationary m m 1 queues within the class of stationary m g 1 queues based on the variance curve

harmonic functions of subordinate killed brownian motion in this paper we study harmonic functions of subordinate killed brownian motion in a domain d we first prove that when the killed brownian semigroup in d is intrinsic ultracontractive all nonnegative harmonic functions of the subordinate killed brownian motion in d are continuous and then we establish a harnack inequality for these harmonic functions we then show that when d is a bounded lipschitz domain both the martin boundary and the minimal martin boundary of the subordinate killed brownian motion in d coincide with the euclidean boundary partial d we also show that when d is a bounded lipschitz domain a boundary harnack principle holds for positive harmonic functions of the subordinate killed brownian motion in d

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

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

bounded minimisers of double phase variational integrals bounded minimisers of the functional w mapsto int dw p a x dw q rm d x where 0 leqq a cdot in c 0 alpha and 1 p q are c 1 beta regular provided the sharp bound q leqq p alpha holds

multiple solutions for a class of double phase problem without the ambrosetti rabinowitz conditions abstract in the present paper in view of the variational approach we consider the existence and multiplicity of weak solutions for a class of the double phase problem div u p 2 u a x u q 2 u f x u in u 0 on where n 2 and 1 p q n firstly by the fountain and dual theorem with cerami condition we obtain some existence of infinitely many solutions for the above problem under some weaker assumptions on f secondly we prove that this problem has at least one nontrivial solution for any parameter 0 small enough and also that the solution blows up in the sobolev norm as 0 finally by imposing additional assumptions on f we establish the existence of infinitely many solutions by using krasnoselskii s genus theory for the above equation

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

existence and uniqueness of elliptic systems with double phase operators and convection terms abstract in this paper we study quasilinear elliptic systems driven by so called double phase operators and nonlinear right hand sides depending on the gradients of the solutions based on the surjectivity result for pseudomonotone operators we prove the existence of at least one weak solution of such systems furthermore under some additional conditions on the data the uniqueness of weak solutions is shown

green function estimates for subordinate brownian motions stable and beyond a subordinate brownian motion x is a l evy process which can be obtained by replacing the time of the brownian motion by an independent subordinator in this paper when the laplace exponent phi of the corresponding subordinator satisfies some mild conditions we first prove the scale invariant boundary harnack inequality for x on arbitrary open sets then we give an explicit form of sharp two sided estimates of the green functions of these subordinate brownian motions in any bounded c 1 1 open set as a consequence we prove the boundary harnack inequality for x on any c 1 1 open set with explicit decay rate unlike cite ksv2 ksv4 our results cover geometric stable processes and relativistic geometric stable process i e the cases when the subordinator has the laplace exponent phi lambda log 1 lambda alpha 2 0 alpha and phi lambda log 1 lambda m alpha 2 2 alpha m 0 0 d 2

existence of solutions for double phase obstacle problems with multivalued convection term abstract the main goal of this paper is the study of an elliptic obstacle problem with a double phase phenomena and a multivalued reaction term which also depends on the gradient of the solution such term is called multivalued convection term under quite general assumptions on the data we prove that the set of weak solutions to our problem is nonempty bounded and closed our proof is based on a surjectivity theorem for multivalued mappings generated by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone mapping

perturbation realization potentials and sensitivity analysis of markov processes two fundamental concepts and quantities realization factors and performance potentials are introduced for markov processes the relations among these two quantities and the group inverse of the infinitesimal generator are studied it is shown that the sensitivity of the steady state performance with respect to the change of the infinitesimal generator can be easily calculated by using either of these three quantities and that these quantities can be estimated by analyzing a single sample path of a markov process based on these results algorithms for estimating performance sensitivities on a single sample path of a markov process can be proposed the potentials in this paper are defined through realization factors and are shown to be the same as those defined by poisson equations the results provide a uniform framework of perturbation realization for infinitesimal perturbation analysis ipa and non ipa approaches to the sensitivity analysis of steady state performance they also provide a theoretical background for the pa algorithms developed in recent years

drift transforms and green function estimates for discontinuous processes in this paper we consider girsanov transforms of pure jump type for discontinuous markov processes we show that under some quite natural conditions the green functions of the girsanov transformed process are comparable to those of the original process as an application of the general results the drift transform of symmetric stable processes is studied in detail in particular we show that the relativistic a stable process in a bounded c 1 1 smooth open set d can be obtained from symmetric a stable process in d through a combination of a pure jump girsanov transform and a feynman kac transform from this we deduce that the green functions for these two processes in d are comparable

nonlinear elliptic equations with variable exponent old and new abstract in this survey paper by using variational methods we are concerned with the qualitative analysis of solutions to nonlinear elliptic problems of the type div a x u u q x 2 u in u 0 on where is a bounded or an exterior domain of r n and q is a continuous positive function the results presented in this paper extend several contributions concerning the lane emden equation and we focus on new phenomena which are due to the presence of variable exponents

dirichlet heat kernel estimates for rotationally symmetric levy processes in this paper we consider a large class of purely discontinuous rotationally symmetric levy processes we establish sharp two sided estimates for the transition densities of such processes killed upon leaving an open set d w hend is a fat open set the sharp two sided estimates are given in terms of surviving probabilities and the global transition density of the levy process when d is a c 1 1 open set and the levy exponent of the process is given by 2 with being a complete bernstein function satisfying a mild growth condition at infinity our two sided estimates are explicit in terms of the distance function to the boundary of d and the levy density of x this gives an affirmative answer to the conjecture posted in chen kim and song global heat kernel estimates for relativistic stable processes in half space like open sets potential anal 36 2012 235 261 our results are the first sharp two sided dirichlet heat kernel estimates for a large class of symmetric levy processes with general levy exponents we also derive an explicit lower bound estimate for symmetric levy processes on r d in terms of their levy exponents

density of smooth functions in wk p x ω kovacik rakosnik investigated the spaces l p x omega of functions which are integrable with variable power p x and the corresponding counterparts of the sobolev spaces w k p x omega we continue that investigation and describe a class of functions p x for which the set of smooth functions on omega is dense in w k p x omega as a corollary we obtain in terms of the distance function a condition on elements of w k p x omega sufficient to ensure that they belong to w 0 k p x omega

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

censored stable processes we present several constructions of a censored stable process in an open set d rn i e a symmetric stable process which is not allowed to jump outside d we address the question of whether the process will approach the boundary of d in a finite time we give sharp conditions for such approach in terms of the stability index and the thickness of the boundary as a corollary new results are obtained concerning besov spaces on non smooth domains including the critical exponent case we also study the decay rate of the corresponding harmonic functions which vanish on a part of the boundary we derive a boundary harnack principle in c1 1 open sets

existence of periodic solutions for ordinary differential systems in musielak orlicz sobolev spaces abstract in this paper we study the periodic solutions for the ordinary differential systems in musielak orlicz sobolev spaces we get some propositions of the musielak orlicz sobolev space w t 1 which are used to get the existence results using variational methods we have some existence results of the periodic solutions for the ordinary differential systems in musielak orlicz sobolev spaces