on mordell weil groups of isotrivial abelian varieties over function fields we show that the mordell weil rank of an isotrivial abelian variety with cyclic holonomy depends only on the fundamental group of the complement to the discriminant provided the discriminant has singularities in cm class introduced here this class of singularities includes all unibranched plane curves singularities as a corollary we describe a family of simple jacobians over the field of rational functions in two variables for which the mordell weil rank is arbitrarily large

theory of differential forms on manifolds in this paper we establish a hodge type decomposition for the lp space of differential forms on closed i e compact oriented smooth rieman nian manifolds critical to the proof of this result is establishing an lp es timate which contains as a special case the l2 result referred to by morrey as gaffney s inequality this inequality helps us show the equivalence of the usual definition of sobolev space with a more geometric formulation which we provide in the case of differential forms on manifolds we also prove the lp boundedness of green s operator which we use in developing the lp theory of the hodge decomposition for the calculus of variations we rigorously verify that the spaces of exact and coexact forms are closed in the lp norm for nonlinear analysis we demonstrate the existence and uniqueness of a solution to the a harmonic equation

graph colourings and partitions in this paper we have investigated mainly the three colouring parameters of a graph g viz the chromatic number the achromatic number and the pseudoachromatic number the importance of their study in connection with the computational complexity partitions algebra projective plane geometry and analysis were briefly surveyed some new results were found along these directions we have rede0ned the concept of perfect graphs in terms of these parameters and obtained a few results some open problems are raised

a threshold of ln n for approximating set cover given a collection f of subsets of s 1 n setcover is the problem of selecting as few as possiblesubsets from f such that their union covers s and maxk cover is the problem of selecting k subsets from f such that their union has maximum cardinality both these problems arenp hard we prove that 1 o 1 ln n is a threshold below which setcover cannot be approximated efficiently unless np has slightlysuperpolynomial time algorithms this closes the gap up to low orderterms between the ratio of approximation achievable by the greedyalogorithm which is 1 o 1 lnn and provious results of lund and yanakakis that showed hardness ofapproximation within a ratio of log 2 n 2s0 72 ln n for max k cover we show an approximationthreshold of 1 1 e up tolow order terms under assumption that p np

global weighted estimates for quasilinear elliptic equations with non standard growth abstract in this paper we obtain a new global gradient estimates in weighted lorentz spaces for weak solutions of p x laplacian type equation with small bmo coefficients in a reifenberg flat domain the modified vitali covering lemma the maximal function technique and the appropriate localization method are the main analytical tools our results improve the known results for such equations

integral estimates for null lagrangians sobolev spaces of differential forms are studied lp projections onto exact forms are introduced as a tool to obtain integral estimates for null lagrangians new results on compensated compactness are given and mean coercive variational integrals are found existence of minima of certain mean coercive functionals is established

the monotonic property of ls μ averaging domains and weighted weak reverse holder inequality abstract we first prove a local weighted weak reverse holder inequality for a harmonic tensors then we study the monotonic property of newly introduced l s averaging domains which can be viewed as an application of the local weighted reverse holder inequality in l s averaging domains by applying the local result we also obtain some similar results in other kinds of domains

semicontinuity problems in the calculus of variations soit f r 3 r une fonction deux fois continument differentiable et pour u w 1 1 a b f u a b a b f x u x u x dx si f r n r m r nm r satisfait 1 f est mesurable en x continue en cs et 2 0 f x s a x c s p p ou p 1 a est une fonction localement sommable non negative et c une constante non negative alors la fonctionnelle f u a b est sequentiellement faiblement semi continue inferieurement sur w 1 p r m si et seulement si f x s est quasi convexe pour chaque s et presque chaque x

the np completeness of some edge partition problems we show that for each fixed n geqq 3 it is np complete to determine whether an arbitrary graph can be edge partitioned into subgraphs isomorphic to the complete graph k n the np completeness of a number of other edge partition problems follows immediately

quasiregular mappings in even dimensions o introduction 1 notation 2 some exterior algebra 3 differential forms in lnm i2 a n 4 differential systems for quasiregular mappings 5 liouville theorem in even dimensions 6 hodge theory in lp r n 7 the beltrami equation in even dimensions 8 the beurling ahlfors operator 9 regularity theorems for quasiregular mappings 10 the caccioppoli type estimate 11 removability theorems for quasiregular mappings 12 some examples appendix the 4 dimensional case

optimal switching for ordinary differential equations we consider the problem of controlling an ordinary differential equation subject to positive switching costs and show in particular that the value functions form the viscosity solution cf 6 7 of the dynamic programming quasi variational inequalities this interpretation allows for a rigorous application of various dynamic programming techniques

general existence theorems for hamilton jacobi equations in the scalar and vectorial cases keywords dirichlet problem hamilton jacobi equation existence of solutions reference caa article 1997 002doi 10 1007 bf02392708view record in web of science record created on 2008 11 25 modified on 2017 05 12

nonlinear cauchy riemann operators in ℝⁿ this paper has arisen from an effort to provide a comprehensive and unifying development of the l p theory of quasiconformal mappings in r n the governing equations for these mappings form nonlinear differential systems of the first order analogous in many respects to the cauchy riemann equations in the complex plane this approach demands that one must work out certain variational integrals involving the jacobian determinant guided by such integrals we introduce two nonlinear differential operators denoted by d and d which act on weakly differentiable deformations f r n of a domain c r n solutions to the so called cauchy riemann equations d f 0 and d f 0 are simply conformal deformations preserving and reversing orientation respectively these operators though genuinely nonlinear possess the important feature of being rank one convex among the many desirable properties we give the fundamental l p estimate d f p a p n d f p in quest of the best constant a p n we are faced with fascinating problems regarding quasiconvexity of some related variational functionals applications to quasiconformal mappings are indicated

existence results for some nonuniformly elliptic equations with irregular data abstract in this paper we prove the existence of solutions of some nonlinear elliptic equations with degenerate coercivity of the type formula when the datum f belongs to the space lr where r 2 and 0 1

global higher integrability for the gradient of very weak solutions of a class of nonlinear elliptic systems abstract the global higher integrability for the gradient of very weak solutions of nonlinear elliptic systems div a x u du b x u du u u 0 w 0 1 r is obtained where is a bounded domain in rn with a lipschitz boundary max 1 p 1

a baire s category method for the dirichlet problem of quasiregular mappings we adopt the idea of baire s category method as presented in a series of papers by dacorogna and marcellini to study the boundary value problem for quasiregular mappings in space our main result is to prove that for any 0 and any piece wise affine map w 1 n r n with d x n ldet d x for almost every x there exists a map u w 1 n r n such that du x n l det du x a e x u u l n the theorems of dacorogna and marcellini do not directly apply to our result since the involved sets are unbounded our proof is elementary and does not require any notion of polyconvexity quasiconvexity or rank one convexity in the vectorial calculus of variations as required in the papers by the quoted authors

global gradient estimates for spherical quasi minimizers of integral functionals with p x growth abstract we investigate spherical quasi minimizers or so called spherical q minimizers u for integral functionals with p x growth of f u f x d u g p x 2 g d u d x in particular we find minimal regularity requirements on the integrand f x the variable exponent p x the boundary of the bounded domain as well as q under which g p l q d u p l q for every q 1 with the desired global calderon zygmund estimate

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

ls μ averaging domains abstract in this paper we first introduce l s averaging domains which are generalizations of l s averaging domains introduced by s g staples we characterize l s averaging domains using the quasihyperbolic metric as applications we prove norm inequalities for conjugate a harmonic tensors in l s averaging domains which can be considered as generalizations of the hardy and littlewood theorem for conjugate harmonic functions finally we give applications to quasiconformal and quasiregular mappings

weighted poincare type inequalities for differential forms inls μ averaging domains abstract we first introduce a new class of weighted functions and obtain some basic properties of this class then as applications of this class we prove local weighted integral inequalities for differential forms finally we obtain the global weighted integral inequalities for differential forms inls averaging domains which can be considered as generalizations of the poincare inequality for sobolev functions

radial weak solutions for the perona malik equation as a differential inclusion abstract the perona malik equation is an ill posed forward backward parabolic equation with some application in image processing in this paper we study the perona malik type equation on a ball in an arbitrary dimension n and show that there exist infinitely many radial weak solutions to the homogeneous neumann boundary problem for smooth nonconstant radially symmetric initial data our approach is to reformulate the n dimensional equation into a one dimensional equation to convert the one dimensional problem into an inhomogeneous partial differential inclusion problem and to apply a baire s category method to the differential inclusion to generate infinitely many solutions

on very weak solutions to a class of double obstacle problems abstract we prove the uniqueness of very weak solutions by hodge decomposition and obtain the existence by an approximation argument moreover we prove that the solution quasiminimizes the r dirichlet integral in k r and finally the stability of very weak solutions is established for varying obstacle functions

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

rainbow graph splitting abstract given an integer c an edge colored graph g is said to be rainbow c splittable if it can be decomposed into at most c vertex disjoint monochromatic induced subgraphs of distinct colors we provide a polynomial time algorithm for deciding whether an edge colored complete graph is rainbow c splittable for not necessarily complete graphs we show that the problem is polynomial if c 2 whereas for c 3 it is np complete even if the graph has maximum degree 2 c 1 finally it remains np complete even for 2 edge colored graphs of maximum degree 7 c 14

calderon zygmund estimates for higher order systems with p x growth for weak solutions u in w m 1 omega mathbb r n of higher order systems of the type int omega left dx int omega left dx for all phi in c infty c omega mathbb r n m 1 with variable growth exponent p 1 we prove that if f p cdot in l q loc omega with 1 0 and in the degenerate case 0

cuspidal plane curves syzygies and a bound on the mw rank abstract let c z f be a reduced plane curve of degree 6k with only nodes and ordinary cusps as singularities let i be the ideal of the points where c has a cusp let s b i s a i s s i be a minimal resolution of i we show that b i 5 k from this we obtain that the mordell weil rank of the elliptic threefold w y 2 x 3 f equals 2 i b i 5 k using this we find an upper bound for the mordell weil rank of w which is 1 18 125 73 2302 106 73 k l o t and we find an upper bound for the exponent of t 2 t 1 in the alexander polynomial of c which is 1 36 125 73 2302 106 73 k l o t this improves a recent bound of cogolludo and libgober almost by a factor 2

the singular set of minima of integral functionals in this paper we provide upper bounds for the hausdorff dimension of the singular set of minima of general variational integrals open image in new window where f is suitably convex with respect to dv and holder continuous with respect to x v in particular we prove that the hausdorff dimension of the singular set is always strictly less than n where open image in new window

on rank one convex and polyconvex conformal energy functions with slow growth we make some remarks about rank one convex and polyconvex functions on the set of all real n n matrices that vanish on the subset k n consisting of all conformal matrices and grow like a power function at infinity we prove that every non negative rank one convex function that vanishes on k n and grows below a power of degree n 2 must vanish identically in odd dimensions n 3 we prove that every non negative polyconvex function that vanishes on k n must vanish identically if it grows below a power of degree n while in even dimensions such polyconvex functions can exist that also grow like a power of half dimension degree

mappings of finite distortion the degree of regularity this paper investigates the self improving integrability properties of the so called mappings of finite distortion let k x 1 be a measurable function defined on a domain rn n 2 and such that exp k x lloc1 0 we show that there exist two universal constants c1 n c2 n with the following property let f be a mapping in wloc1 1 rn with df x n k x j x f for a e x and such that the jacobian determinant j x f is locally in l1log c1 n l then automatically j x f is locally in l1logc2 n l this result constitutes the appropriate analog for the self improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite distortion namely we obtain novel results on the size of removable singularities for bounded mappings of finite distortion and on the area distortion under this class of mappings

on a reverse estimate for hodge decompositions of p laplacian type operators let x p 2 be a p laplacian type operator and consider the hodge decomposition x du d h div h 0 a standard elliptic theory asserts that d q p 1 c du p 1q for all q p 1 there has been considerable recent interest in the validity of the reverse estimate du p 1q c d q p 1 for q p 1 in the regularity study of certain geometrical mappings in this paper we give a relatively new proof of a well known theorem that this reverse estimate holds for all q sufficiently close to the natural power p and also prove that the estimate holds for all q p 1 for certain special weak solutions u

the p laplace system with right hand side in divergence form inner and up to the boundary pointwise estimates abstract in this note we collect some very recent pointwise bounds for the gradient of solutions and for the solutions themselves to the p laplace system with right hand side in divergence form both estimates inside the domain for local solutions and global estimates for solutions to boundary value problems are discussed their formulation involves sharp maximal operators whose properties enable us to translate some aspects of the elliptic regularity theory into a merely harmonic analytic framework as a consequence a flexible comprehensive approach to estimates for solutions to the p laplace system for a broad class of norms is derived in particular global estimates under minimal boundary regularity are presented

optimal existence theorems for nonhomogeneous differential inclusions abstract in this paper we address the question of solvability of the differential inclusions 1 1 our approach to these problems is based on the idea of constructing a sequence of approximate solutions which converges strongly and makes use of gromov s idea following earlier work of nash and kuiper to control convergence of the gradients by appropriate selection of the elements of the sequence in this paper we identify an optimal setting of this method in particular we show that the existence result holds for general upper semicontinuous functions h without extra requirements like quasiconvexity of h with respect to du which was assumed in previous works where the idea to apply the baire category lemma to the sets of approximate solutions was developed we also apply our result to identify the minimal sets where the function h should vanish to guarantee solvability of the inclusions

global gradient estimates for a borderline case of double phase problems with measure data abstract we obtain a global calderon zygmund estimate for a borderline case of double phase problems with measure data in terms of the 1 fractional maximal function of the measure

calderon zygmund type estimates for a class of obstacle problems with p x growth for local minimizers u wloc1 p of quasiconvex integral functionals of the type f u f x du x dx with p x growth in the class k u wloc1 p u where wloc1 p is a given obstacle function we show estimates of calderon zygmund type i e d p llocq du p llocq for any q 1 provided that the modulus of continuity of the exponent function p satisfies the condition log1 0as 0

rank one convexity does not imply quasiconvexity we consider variational integrals defined for sufficiently regular functions u r m here is a bounded open subset of r n du x denotes the gradient matrix of u at x and f is a continuous function on the space of all real m n matrices m m n one of the important problems in the calculus of variations is to characterise the functions f for which the integral i is lower semicontinuous in this connection the following notions were introduced see 3 9 10

quasiconformal 4 manifolds for any pseudo group of homeomorphism of euclidean space one can define the corresponding category of manifolds the most familiar examples in topology are the full pseudo group of homeomorphisms giving rise to the theory of topological manifolds and the subgroup of smooth differomorphisms giving rise to the theory of c manifolds in this paper we discuss an intermediate category quasiconformal homeomorphisms and manifolds recall that a homeomorphism 9 d r n from a domain d in r n to its image 9 d is k quasiconformal if for all x in d

global integrability theorems for a harmonic tensors abstract we extend global integrability theorems for the gradients of a harmonic functions to the exterior derivative of differential forms satisfying rather general nonhomogeneous elliptic equations these include the usual a harmonic equations geometric conditions on the boundary of the domains of integration imply a corresponding exponent of integrability in the process we generalize the weak reverse holder inequality to such differential forms

the liouville theorem under second order differentiability assumption abstract iwaniec and martin proved that in even dimensions n 3 w loc 1 n 2 conformal mappings are mobius transformations and they conjectured that it should also be true in odd dimensions we prove this theorem for a conformal map f w loc 1 1 in dimension n 3 under one additional assumption that the norm of the first order derivative d f satisfies d f p w loc 1 2 for p n 2 4 this is optimal in the sense that if d f p w loc 1 2 for p n 2 4 it may not be a mobius transform this result shows the necessity of the sobolev exponent in the iwaniec martin conjecture meanwhile we show that the iwaniec martin conjecture can be reduced to a conjecture about the caccioppoli type estimate

on the minimum monochromatic or multicolored subgraph partition problems let g v e be an edge colored graph a subgraph h is said to be monochromatic if all the edges of h have the same color and multicolored if no two edges of h have the same color we investigate the complexity of the problems for finding the minimum number of monochromatic or multicolored subgraphs such as cliques cycles trees and paths partitioning v g depending on the number of colors used and the maximal number of times a color appears in a coloring we also present a greedy scheme that yields a lnm 1 approximation for the problem of finding the minimum number of monochromatic cliques partitioning v g for a k 4 free graph g where m is the size of the largest monochromatic clique in g by a slightly modification of the approximation algorithm it can be used for the multicolored case we show that unless np dtime n o l o g l o g n for any e 0 there is no approximation algorithm for finding the minimum number of multicolored trees partitioning v g with performance 50 521 1 e ln v

nonlinear commutators for the fractional p laplacian and applications we prove a nonlinear commutator estimate concerning the transfer of derivatives onto testfunctions for the fractional p laplacian this implies that solutions to certain degenerate nonlocal equations are higher differentiable also weakly fractional p harmonic functions which a priori are less regular than variational solutions are in fact classical as an application we show that sequences of uniformly bounded frac n s harmonic maps converge strongly outside at most finitely many points

weighted integral inequalities for conjugate a harmonic tensors abstract we first prove a local weighted integral inequality for conjugate a harmonic tensors then as an application of our local result we prove a global weighted integral inequality for conjugate a harmonic tensors in ls averaging domains which can be considered as a generalization of the classical result finally we give applications of the above results to quasiregular mappings

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

cauchy dirichlet problem for first order nonlinear systems abstract let r nbe open u r mand thus the gradient matrixdu r m n we lete r m nbe compact and denote byrcoeandpcoethe rank one convex and polyconvex hull ofe respectively we show that ifrcoe pcoe and two other hypotheses named the segment property and the extreme points property and if c1 r m is such that d x e int pcoe x then there exists a dense set of u w1 r m such that formula we apply this existence theorem to some relevant examples studied in the literature as well as to problems with x u dependence

boundary regularity in variational problems we prove that if u omega subset mathbb r n to mathbb r n is a solution to the dirichlet variational problem mathop rm min limits w int omega f x w dw rm d x quad rm subject to quad w equiv u 0 rm on partial omega involving a regular boundary datum u 0 and a regular integrand f x w dw strongly convex in dw and satisfying suitable growth conditions then mathcal h n 1 almost every boundary point is regular for u in the sense that du is holder continuous in a relative neighborhood of the point the existence of even one such regular boundary point was previously not known except for some very special cases treated by jost meier math ann 262 549 561 1983 our results are consequences of new up to the boundary higher differentiability results that we establish for minima of the functionals in question the methods also allow us to improve the known boundary regularity results for solutions to non linear elliptic systems and in some cases to improve the known interior singular sets estimates for minimizers moreover our approach allows for a treatment of systems and functionals with rough coefficients belonging to suitable sobolev spaces of fractional order

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

some examples of conjugatep harmonic differential forms abstract there has been new interest developing in the study of the hodge theory and p harmonic differential forms in recent years largely pertaining to applications in quasi conformal analysis electromagnetic theory relativity theory mathematical physics nonlinear potential theory etc in this paper we develop an effective method to find conjugate p harmonic differential forms in r n by applying hodge theory in calculations with the p harmonic equation as applications of the method we obtain some interesting examples of conjugate p harmonic differential forms

generalized poincare inequalities for solutions to the a harmonic equation in certain domains abstract we first prove local versions of the poincare inequality for solutions to the a harmonic equation then as applications of the local results we obtain the global versions of the poincare inequality for solutions to the a harmonic equation in l s 0 averaging domains and l s averaging domains respectively these results can be considered as generalizations of the classical poincare theorem

regularity in oscillatory nonlinear elliptic systems we give uniform bounds in morrey sobolev spaces h1 for weak solutions ue to oscillatory nonlinear elliptic systems the main novelty is in the precise qualitative description of the e uniform morrey exponent in terms of the ellipticity ratio of the elliptic systems this is achieved by showing that weak solutions of general nonlinear elliptic systems with lipschitz coefficients are of class w2 p locally or globally depending on assumptions for an exponent p 2 that is independent of dimensions and for which p 2 is inversely proportional to the ellipticity ratio