solvable groups acting on the line actions of discrete groups on the real line are considered when the group of homeomorphisms is solvable several sufficient conditions are given which guarantee that the group is semiconjugate to a subgroup of the affine group of the line in the process of obtaining these results sufficient conditions are also determined for the existence of invariant quasi invariant measures for abelian solvable groups acting on the line it turns out for example that any solvable group of real analytic diffeomorphisms or a polycyclic group of homeomorphisms has a quasi invariant measure and that any abelian group of c diffeomorphisms has an invariant measure an example is given to show that some restrictions are necessary in order to obtain such conclusions some applications to the study of codimension one foliations are indicated introduction if g is a group of homeomorphisms of the line it is reasonable to ask for a dynamic description of how g acts when reasonable restrictions are placed on g for example in 9 it is shown that if g is finitely generated and nilpotent then there is a g invariant borel measure on r which is finite on compact sets this already says much about g the proof is quite different from classical arguments which guarantee the existence of a finite invariant measure when an amenable group acts on a compact hausdorff space 6 in that one uses pseudogroup properties of g rather than group properties specifically nonexponential growth of orbits rather than nilpotence of g it is reasonable to determine what can be said when g is solvable but as pointed out in 9 solvable groups need not have invariant measures e g subgroups of the affine group which contain nontrivial translations and nontrivial dilations such examples suggest consideration of a more general invariance property a measure is called quasi invariant for g if each element of g multiplies the measure by a nonzero constant which depends on the particular element of g usual lebesgue measure is quasi invariant for the affine group so it is reasonable to ask if every finitely generated solvable g has a quasi invariant measure it turns out that a large class of solvable groups including polycyclic groups acting on the line must have quasi invariant measures but there are solvable groups which do not have such measures an important step in the investigation of solvable groups is the consideration of infinitely generated abelian groups if differentiability is assumed stronger received by the editors october 30 1980 and in revised form august 3 1982 1980 mathematics subject classification primary 57e25 secondary 58f11 58f15 58f18

optimal designs for bivariate logistic regression summary in drug testing experiments the primary responses of interest are efficacy and toxicity these can be modeled as a bivariate quantal response using the gumbel model for bivariate logistic regression d optimal and q optimal experimental designs are developed for this model the q optimal design minimizes the average asymptotic prediction variance of p l 0 d the probability of efficacy without toxicity at dose d over a desired range of doses the robustness of these designs to parameter misspecification is discussed in addition d efficiencies of q optimal designs and q efficiencies of doptimal designs are presented an extension of the general equivalence theorem to the multivariate case is applied to these designs

regulating flows topology of foliations and rigidity a flow transverse to a foliation is regulating if in the universal cover an arbitrary orbit of the flow intersects every leaf of the lifted foliation this implies that the foliation is r covered that is its leaf space in the universal cover is homeomorphic to the reals we analyse the converse of this implication to study the topology of the leaf space of certain foliations we prove that if a pseudo anosov flow is transverse to an r covered foliation and the flow is not an r covered anosov flow then the flow is regulating for the foliation using this we show that several interesting classes of foliations are not r covered finally we show a rigidity result if an r covered anosov flow is transverse to a foliation but is not regulating then the foliation blows down to one topologically conjugate to the stable or unstable foliations of the transverse flow

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

adaptive optimization and d optimum experimental design we consider the situation where one has to maximize a function eta theta mathbf x with respect to mathbf x epsilon mathbb r q when theta is unknown and estimated by least squares through observations y k mathbf f top mathbf x k theta varepsilon k with varepsilon k some random error classical applications are regulation and extremum control problems the approach we adopt corresponds to maximizing the sum of the current estimated objective and a penalization for poor estimation mathbf x k 1 maximizes eta hat theta k mathbf x alpha k k d k mathbf x with hat theta k the estimated value of theta at step k and d k the penalization function sufficient conditions for strong consistency of hat theta k and for almost sure convergence of 1 k sigma i 1 k eta theta mathbf x i to the maximum value of eta theta mathbf x are derived in the case where d k cdot is the variance function used in the sequential construction of d optimum designs a classical sequential scheme from adaptive control is shown not to satisfy these conditions and numerical simulations confirm that it indeed has convergence problems

a regularity theorem for minimizers of quasiconvex integrals we prove c1 partial regularity for minimizers of functionals with quasiconvex integrand f x u du depending on vector valued functions u the integrand is required to be twice continuously differentiable in du and no assumption on the growth of the derivatives of f is made a polynomial growth is required only on f itself

harnack inequality for quasilinear elliptic equations on riemannian manifolds abstract we study viscosity solutions to degenerate and singular elliptic equations div f u u u h of p laplacian type on riemannian manifolds where an even function f c 1 r c 2 0 is supposed to be strictly convex on 0 under the assumption that either f c 2 r or its convex conjugate f c 2 r with some structural condition we establish a locally uniform abp type estimate and the krylov safonov type harnack inequality on riemannian manifolds with the use of an intrinsic geometric quantity to the operator here the c 2 regularities of f and f account for degenerate and singular operators respectively

regularity for minimizers of integrals with nonstandard growth abstract we deal with variational integrals f x d u x d x and we consider a minimizer u r n r among all functions that agree on the boundary with some fixed boundary value u we assume that the boundary datum u makes the density f x d u x more integrable and we prove that the minimizer u enjoy higher integrability

quasigeodesic pseudo anosov flows in hyperbolic 3 manifolds and connections with large scale geometry abstract in this article we obtain a simple topological and dynamical systems condition which is necessary and sufficient for an arbitrary pseudo anosov flow in a closed hyperbolic three manifold to be quasigeodesic quasigeodesic means that orbits are efficient in measuring length up to a bounded multiplicative distortion when lifted to the universal cover we prove that such flows are quasigeodesic if and only if there is an upper bound depending only on the flow on the number of orbits which are freely homotopic to an arbitrary closed orbit of the flow the main ingredient is a proof that under the boundedness condition the fundamental group of the manifold acts as a uniform convergence group on a flow ideal boundary of the universal cover we also construct a flow ideal compactification of the universal cover and prove that it is equivariantly homeomorphic to the gromov compactification this implies the quasigeodesic behavior of the flow the flow ideal boundary and flow ideal compactification are constructed using only the structure of the flow

finding nicer mappings with lower or equal energy in nonlinear elasticity abstract in the framework of nonlinear elasticity we consider the model energy f u d u x p h det d u x d x where u r n r n with det d u 0 and h 0 0 is convex moreover h t blows up when t 0 we study the problem fix the mapping u with finite energy f u and find a mapping v with the same boundary values with det d v 0 and energy f v not higher than f u such that every component v of v enjoys a kind of maximum and minimum principle due to the constraint det d v 0 a truncation argument does not work on the contrary the constraint det d v 0 makes the problem easy when p n since it is known that every component is weakly monotone so we can take v u in the present work we address the case 2 p n and we give a positive answer if the blow up of h t is of logarithmic type

regularity results for a class of non autonomous obstacle problems with p q growth abstract we study some regularity issues for solutions of non autonomous obstacle problems with p q growth under suitable assumptions our analysis covers the main models available in the literature

anosov flows surface groups and curves in projective space note that in 10 w goldman gives a complete description of these connected components in the case of finite covers of psl 2 r in the case of psl 2 r two homeomorphic components called teichmuller spaces play a central role these two components are well known to be homeomorphic to a ball of dimension 6g 6 n hitchin generalises this situation to psl n r indeed one of these components when n is odd and two when n is even has a very simple topology let us define an n fuchsian representation to be a representation which can be written as 0 where 0 is a cocompact representation with values in psl 2 r and is the irreducible representation of psl 2 r in psl n r we denote by reph 1 s psl n r a connected component that contains fuchsian representations and call it a hitchin component in fact when n is odd there is one hitchin component and when n is even two isomorphic ones n hitchin proves in 17

a compact embedding result for anisotropic sobolev spaces associated to a strip like domain and some applications abstract let m 1 and d 2 be integers and consider a strip like domain o r d where o r m is a bounded euclidean domain with smooth boundary furthermore let p o r d r be a uniformly continuous and cylindrically symmetric function we prove that the subspace of w 1 p x y o r d consisting of the cylindrically symmetric functions is compactly embedded into l o r d provided that m d p inf x y o r d p x y p sup x y o r d p x y as an application we study a neumann problem involving the p x y laplacian operator and an oscillating nonlinearity proving the existence of infinitely many cylindrically symmetric weak solutions our approach is based on variational and topological methods in addition to the principle of symmetric criticality

foliations with good geometry the goal of this article is to show that there is a large class of closed hyperbolic 3 manifolds admitting codimension one foliations with good large scale geometric properties we obtain results in two directions first there is an internal result a possibly singular foliation in a manifold is quasi isometric if when lifted to the universal cover distance along leaves is efficienit up to a bounded multiplicative distortion in measuring distance in the universal cover this means that leaves reflect very well the geometry in the large of the universal cover and are geometrically tight this is the best geometric behavior we previously proved that nonsingular codimension one foliations in closed hyperbolic 3 manifolds can never be quasi isometric in this article we produce a large class of singular quasi isometric codimension one foliations in closed hyperbolic 3 manifolds the foliations are stable and unstable foliations of pseudo anosov flows our second result is an external result relating nonsingular foliations in hyperbolic 3 manifolds with their limit sets in the universal cover that is showing that leaves in the universal cover have good asymptotic behavior let g be a reebless finite depth foliation in a closed hyperbolic 3 manifold then g is not quasi isometric but suppose that g is transverse to a quasigeodesic pseudo anosov flow with quasi isometric stable and unstable foliations which are given by the internal result we then show that the lifts of leaves of g to the universal cover extend continuously to the sphere at infinity and we also produce infinitely many examples satisfying the hypothesis the main tools used to prove these results are first a link between geometric properties of stable unstable foliations of pseudo anosov flows and the topology of these foliations in the tuniversal cover and second a topological theory of the joint structure of the pseudo anosov foliations in the universal cover reebless codimension one foliations are extremely useful for understanding the topology of 3 manifolds for instance they imply that the manifold is irreducible ro its universal cover is homeomorphic to r3 pa leaves are 7rl injective no and transversals are never null homotopic no hence they reflect topological properties of the manifold as for which manifolds have reebless foliations gabai gal ga2 ga3 proved that any compact oriented irreducible 3 manifold with nonzero first betti number has many reebless finite depth foliations roughly a

harmonic type approximation lemmas we give a survey of known and not known harmonic type approximation lemmas which are descendants of the classical de giorgi s one and we outline some of their recent or possible applications

existence of parabolic minimizers on metric measure spaces abstract the objects of our studies are vector valued parabolic minimizers u associated to a convex caratheodory integrand f obeying a p growth assumption from below and a certain monotonicity condition in the gradient variable here the functions being considered are defined on a metric measure space x d for such parabolic minimizers that coincide with a time independent cauchy dirichlet datum u 0 on the parabolic boundary of a space time cylinder 0 t with an open subset x and t 0 we prove existence in the parabolic newtonian space l p 0 t n 1 p r n in this paper we generalize results from bogelein et al 2014 2015 to the metric setting and argue completely on a variational level

higher integrability for minimizers of integral functionals with p q growth abstract we prove higher integrability for minimizers u r n of integral functionals f du a x u dx where f satisfies a non standard growth condition of p q type z p f z l 1 z q p

promoting essential laminations we show that a co orientable taut foliation of a closed orientable algebraically atoroidal 3 manifold is either the weak stable foliation of an anosov flow or else there are a pair of very full genuine laminations transverse to the foliation

regularity results for vectorial minimizers of a class of degenerate convex integrals abstract we establish the higher differentiability and the higher integrability for the gradient of vectorial minimizers of integral functionals with p q growth conditions we assume that the non homogeneous densities are uniformly convex and have a radial structure with respect to the gradient variable only at infinity the results are obtained under a possibly discontinuous dependence on the spatial variable of the integrand

two stage design for dose finding that accounts for both efficacy and safety we introduce a two stage design for dose finding in the context of phase i ii studies where two binary correlated endpoints are available for instance one for efficacy and one for toxicity the bivariate probit model is used as a working model for the dose response relationship given a desirable point for the marginal probabilities of efficacy and toxicity the goal is to find the target dose that is closest to the desirable point the criterion of optimality objective function is the variance of the estimator for that dose optimal experimental design methodology is used to construct efficient dose allocation procedures that treat patients in the study at doses that are both safe and efficacious

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

existence of evolutionary variational solutions via the calculus of variations abstract in this paper we introduce a purely variational approach to time dependent problems yielding the existence of global parabolic minimizers that is 0 t u t f x d u d x d t 0 t f x d u d d x d t whenever t 0 and c 0 0 t r n for the integrand f r n n 0 we merely assume convexity with respect to the gradient variable and coercivity these evolutionary variational solutions are obtained as limits of maps depending on space and time minimizing certain convex variational functionals in the simplest situation with some growth conditions on f the method provides the existence of global weak solutions to cauchy dirichlet problems of parabolic systems of the type t u div d f x d u 0 in 0

on l spaces and left orderable fundamental groups examples suggest that there is a correspondence between l spaces and three manifolds whose fundamental groups cannot be left ordered in this paper we establish the equivalence of these conditions for several large classes of manifolds in particular we prove that they are equivalent for any closed connected orientable geometric three manifold that is non hyperbolic a family which includes all closed connected orientable seifert fibred spaces we also show that they are equivalent for the twofold branched covers of non split alternating links to do this we prove that the fundamental group of the twofold branched cover of an alternating link is left orderable if and only if it is a trivial link with two or more components we also show that this places strong restrictions on the representations of the fundamental group of an alternating knot complement with values in text homeo s 1

optimal dose finding designs with correlated continuous and discrete responses in dose finding clinical studies it is common that multiple endpoints are of interest for instance in phase i ii studies efficacy and toxicity are often the primary endpoints which are observed simultaneously and which need to be evaluated together motivated by this we confine ourselves to bivariate responses and focus on the most analytically difficult case a mixture of continuous and categorical responses we adopt the bivariate probit dose response model and quantify our goal by a utility function we study locally optimal designs two stage optimal designs and fully adaptive designs under different ethical and cost constraints in the experiments we assess the performance of two stage designs and fully adaptive designs via simulations our simulations suggest that the two stage designs are as efficient as and may be more efficient than the fully adaptive designs if there is a moderate sample size in the initial stage in addition two stage designs are easier to construct and implement and thus can be a useful approach in practice copyright 2011 john wiley sons ltd

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

c1 α local regularity of weak solutions of degenerate elliptic equations abstract it is demonstrated the local c 1 alpha nature of weak solutions of elliptic equations of the type 1 1 in the introduction under the degeneracy or singularity assumptions a sub 1 a sub 3

the boundary of negatively curved groups gromov s article gr contains fundamental properties of negatively curved groups several sets of seminar notes are available frn swn usn that contain more detailed accounts of and further expand on gromov s article a sequence xi e r i 1 2 is convergent at infinity provided xi x t 00 as i j t 00 two sequences xi yj convergent at infinityare equivalent if xi yi t 00 as i t 00 a point in the boundary ar of r is an equivalence class of sequences convergent at infinity now r u ar has a natural topology in which r is a discrete subspace and a typical not necessarily open neighborhood of a point a e ar represented by a sequence xj is given by y e r u arj a y n where a y lim xi y if y e r and a y lim xi yi if year is represented by a sequence yj ar is a compact metrizable finite dimensional space gr swn the important tool that relates ar with the cohomological properties of r is the rips complex pd r for every d 0 pd r is the simplicial complex whose vertices are elements of r and a collection xi xk e r spans a simplex if d xi x d for all i j the natural group action of r on itself by left translations gives rise to an action on pd r the key observation of rips is that when d is sufficiently large pd r is contractible and therefore when r is torsion free provides a model for er more precisely we have proposition 1 1 gr lemma 1 7 a swn 4 2 proposition 9 suppose that r is a negatively curved group and let t5 be as in the definition choose an integer

higher differentiability of minimizers of variational integrals with variable exponents in this paper we study the regularity properties of local minimizers of integral functionals of the form f v f x dv x dx where the integrand f is convex and satisfies the so called p x growth conditions

differentiability for bounded minimizers of some anisotropic integrals abstract we prove the existence of second weak derivatives for bounded minimizers u rn rn of the integral du 2 dnu q dx when 2

regularity results for a class of obstacle problems with nonstandard growth abstract we consider the obstacle problem related to the following energy with nonstandard growth d u p x log e d u d x we investigate the regularity properties of solutions to the obstacle problems along with a suitable assumptions on the variable exponent p and the obstacle

the heat flow in nonlinear hodge theory we study the nonlinear hodge systemd 0 and 2 0 for an exterior form on a compact oriented riemannian manifold m where q is a given positive function the solutions are called harmonic forms they are the stationary points on cohomology classes of the functional e me 2 dm with e q q 2 the codifferential of a form is defined as 1 with 2 we evolve a given closed form 0 by the nonlinear heat flow system d for a time dependent exterior form x t on m this system is the differential of the normalized gradient flow u for e with 0 du under a technical assumption on the function 2 q q q we show that the nonlinear heat flow system d with initial condition 0 0 has a unique solution for all times which converges to a harmonic form in the cohomology class of 0 this yields a nonlinear hodge theorem that every cohomology class of m has a unique harmonic representative

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

existence results to elliptic systems with nonstandard growth conditions abstract nonstandard growth conditions in partial differential equations have been the subject of recent developments in elastic mechanics and electrorheological fluid dynamics lecture notes in mathematics vol 1748 2000 c r acad sci paris ser i math 329 1999 393 398 math ussr izv 29 1987 33 66 in this work elliptic systems with nonstandard growth conditions are studied existence and multiplicity results under growth conditions on the reaction terms are established

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

nil anosov actions we consider anosov actions of a lie group g of dimension k on a closed manifold of dimension k n we introduce the notion of nil anosov action of g which includes the case where g is nilpotent and establishes the invariance by the entire group g of the associated stable and unstable foliations we then prove a spectral decomposition theorem for such an action when the group g is nilpotent finally we focus on the case where g is nilpotent and the unstable bundle has codimension one we prove that in this case the action is a nil extension over an anosov action of an abelian lie group in particular if n ge 3 then the action is topologically transitive if n 2 then the action is a nil extension over an anosov flow

integrability for solutions to some anisotropic elliptic equations abstract we consider the boundary value problem i 1 n d i a i x d u x 0 x u x u x x we show that higher integrability of the boundary datum u forces solutions u to have higher integrability as well assumptions on a i x z are suggested by the euler equation of the anisotropic functional d 1 u p 1 d 2 u p 2 d n u p n d x

higher differentiability for n harmonic systems with sobolev coefficients abstract we provide a higher differentiability result for the finite energy solutions u w 1 1 r n of a degenerate elliptic system of the form div a x d u 0 in r n where is a bounded open set in r n and a x is a degenerate elliptic operator the function which measures the degree of degeneracy of the problem is assumed to belong to the sobolev class w 1 n

a spectral like decomposition for transitive anosov flows in dimension three given a transitive or non transitive anosov vector field x on a closed three dimensional manifold m one may try to decompose m x by cutting m along tori and klein bottles transverse to x we prove that one can find a finite collection s 1 dots s n of pairwise disjoint pairwise non parallel tori and klein bottles transverse to x such that the maximal invariant sets lambda 1 dots lambda m of the connected components v 1 dots v m of m s 1 cup dots cup s n satisfy the following properties each lambda i is a compact invariant locally maximal transitive set for x the collection lambda 1 dots lambda m is canonically attached to the pair m x i e it can be defined independently of the collection of tori and klein bottles s 1 dots s n the lambda i s are the smallest possible for every possibly infinite collection s i i in i of tori and klein bottles transverse to x the lambda i s are contained in the maximal invariant set of m cup i s i to a certain extent the sets lambda 1 dots lambda m are analogs for anosov vector field in dimension 3 of the basic pieces which appear in the spectral decomposition of a non transitive axiom a vector field then we discuss the uniqueness of such a decomposition we prove that the pieces of the decomposition v 1 dots v m equipped with the restriction of the anosov vector field x are almost unique up to topological equivalence

homology of closed geodesics in a negatively curved manifold soit m une variete de riemann compacte dont le flot geodesique sur le fibre tangent unite est de type anosov si on note par n x le nombre de geodesiques premieres p dans m dont la classe d homologie est un donne et de longueur l p x alors le taux de croissance exponentiel lim x x a1 log n x est egal a l entropie topologique h du flot geodesique 0

calderon zygmund estimate for asymptotically regular non uniformly elliptic equations abstract we prove a global calderon zygmund estimate for the distributional solution to non uniformly elliptic equation with an asymptotically regular nonlinearity in a c 1 domain for some 0 here a key ingredient of our proof is to make use of approximating the solution of asymptotically regular problem by the solution of regular problem while the gradient of its solution close to infinity

on invariant volumes of codimension one anosov flows and the verjovsky conjecture we show that any topologically transitive codimension one anosov flow on a closed manifold is topologically equivalent to a smooth anosov flow that preserves a smooth volume by a classical theorem due to verjovsky any higher dimensional codimension one anosov flow is topologically transitive recently simic showed that any higher dimensional codimension one anosov flow that preserves a smooth volume is topologically equivalent to the suspension of an anosov diffeomorphism therefore our result gives a complete classification of codimension one anosov flows up to topological equivalence in higher dimensions

on the lipschitz regularity of minimizers of anisotropic functionals abstract we prove a lipschitz regularity result for minimizers of functionals of the calculus of variations of the form f du x dx where f is a continuous convex function from r n into 0 not necessarily depending on the modulus of du

optimal designs for active controlled dose finding trials with efficacy toxicity outcomes author s schorning kirsten dette holger kettelhake katrin wong weng kee bretz frank abstract we derive optimal designs to estimate efficacy and toxicity in active controlled dose finding trials whenthe bivariate continuous outcomes are described using nonlinear regression models we determine upperbounds on the required number of different doses and provide conditions under which the boundary pointsof the design space are included in the optimal design we provide an analytical description of minimallysupported optimal designs and show that they do not depend on the correlation between the bivariateoutcomes

regularity for the weak solutions to certain parabolic systems under certain growth condition abstract in this paper we consider the regularity of the weak solutions to a quasilinear parabolic systems which is a generalization of p laplacian of the type u t i a i u x f i x t u u i 1 n where the main part satisfies some ellipticity and f i satisfies certain growth conditions we prove boundedness of the solutions and the gradients of solutions to the systems by the means of the energy estimates and a nonlinear iteration procedure of the moser type in this paper

interior behaviour of minimizers for certain functionals with nonstandard growth for all t e 0 00 for example we see that f t2 atp btq a o b o l p 1 satisfy the ellipticity condition 2 recently a number of authors have considered regularity questions of minimizers for z under nonstandard growth conditions onf see l 7 indeed in the scalar cases choe 3 has already obtained c regularity result for minimizers of z when f t2 tp log 1 t p 1 using lewis method 8 in this paper we prove that a locally bounded minimizer u is c f under the assumption that l prq p l 3

global boundedness and holder continuity of quasiminimizers with the general nonstandard growth conditions abstract in this paper we study the integral functionals with the general nonstandard growth we prove the boundedness and holder continuity of quasiminimizers of these functionals our results for quasiminimizers improve variable exponent case and generalize constant exponent case studied in direct methods in the calculus of variations 2003 by giusti

global integrability for weak solutions to some anisotropic elliptic equations abstract we consider the boundary value problem i 1 n d i a i x d u x 0 x u x u x x we show that higher integrability of the boundary datum u forces solutions u to have higher integrability as well assumptions on a i x z are suggested by euler equation of the anisotropic functional i 1 n 2 d i u 2 d i u sin d i u p i 2