least squares mixed finite element methods for non selfadjoint elliptic problems i error estimates a least squares mixed finite element method for general second order non selfadjoint elliptic problems in two and three dimensional domains is formulated and analyzed the finite element spaces for the primary solution approximation u h and the flux approximation sigma h consist of piecewise polynomials of degree k and r respectively the method is mildly nonconforming on the boundary the cases k r and k 1 r are studied it is proved that the method is not subject to the lbb condition optimal l 2 and h 1 error estimates are derived for regular finite element partitions numerical experiments confirming the theoretical rates of convergence are presented

an existence result for superparabolic functions abstract we study superparabolic functions related to nonlinear parabolic equations they are defined by means of a parabolic comparison principle with respect to solutions we show that every superparabolic function satisfies the equation with a positive radon measure on the right hand side and conversely for every finite positive radon measure there exists a superparabolic function which is solution to the corresponding equation with measure data

generalized hyperelliptic surfaces this article presents some results on the surfaces of general type whose albanese morphism is a holomorphic fibre bundle

sobolev preconditioning for the poisson boltzmann equation abstract this paper presents an overview of least squares steepest descent using sobolev gradients for several prototype differential equations in the linear case the method is viewed as a very effective preconditioning strategy for the basic iterative method which arises from steepest descent in particular it acts to smooth the euclidean gradient results are given for the one dimensional possion boltzmann equation from semiconductor device modeling

on a conjecture of voisin on the gonality of very general abelian varieties abstract we study orbits for rational equivalence of zero cycles on very general abelian varieties by adapting a method of voisin to powers of abelian varieties we deduce that for k at least 3 a very general abelian variety of dimension at least 2 k 2 has covering gonality greater than k this settles a conjecture of voisin we also discuss how upper bounds for the dimension of orbits for rational equivalence can be used to provide new lower bounds on other measures of irrationality in particular we obtain a strengthening of the alzati pirola bound on the degree of irrationality of abelian varieties

nonlinear parabolic equations with measure data in this paper we give summability results for the gradients of solutions of nonlinear parabolic equations whose model isu div u p 2 u on 0 t p with homogeneous cauchy dirichlet boundary conditions wherep 1 and is a bounded measure on 0 t we also study how the summability of the gradient improves if the measure is a function inlm 0 t withm small moreover we give a new proof of the existence of a solution for problem p

simplifying stable mappings into the plane from a global viewpoint let f m r2 be a c stable map of an n dimensional manifold into the plane the main purpose of this paper is to define a global surgery operation on f which simplifies the configuration of the critical value set and which does not change the diffeomorphism type of the source manifold m for this purpose we also study the quotient space wf of f which is the space of the connected components of the fibers of f and we completely determine its local structure for arbitrary dimension n of the source manifold m this is a completion of the result of kushner levine and porto for dimension 3 and that of furuya for orientable manifolds of dimension 4 we also pay special attention to dimension 4 and obtain a simplification theorem for stable maps whose regular fiber is a torus or a 2 sphere which is a refinement of a result of kobayashi

bielliptic curves of genus three and the torelli problem for certain elliptic surfaces abstract we study the hodge structure of elliptic surfaces which are canonically defined from bielliptic curves of genus three we prove that the period map for the second cohomology has one dimensional fibers and the period map for the total cohomology is of degree twelve and moreover by adding the information of the hodge structure of the canonical divisor we prove a generic torelli theorem for these elliptic surfaces finally we give explicit examples of the pair of non isomorphic elliptic surfaces which have the same hodge structure on themselves and the same hodge structure on their canonical divisors

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

on symmetric products of curves let c be a smooth complex projective curve of genus g and let c be its second symmetric product this paper concerns the study of some attempts at extending to c the notion of gonality in particular we prove that the degree of irrationality of c is at least g 1 when c is generic and that the minimum gonality of curves through the generic point of c equals the gonality of c in order to produce the main results we deal with correspondences on the k fold symmetric product of c with some interesting linear subspaces of p enjoying a condition of cayley bacharach type and with monodromy of rational maps as an application we also give new bounds on the ample cone of c when c is a generic curve of genus 6 g 8

galois closure and lagrangian varieties abstract let x be a complex projective variety and consider the morphism k k h 0 x x 1 h 0 x x k we use galois closures of finite rational maps to introduce a new method for producing varieties such that k has non trivial kernel we then apply our result to the two dimensional case and we construct a new family of surfaces which are lagrangian in their albanese variety moreover we analyze these surfaces computing their chern invariants and proving that they are not fibred over curves of genus g 2 the topological index of these surfaces is negative and this provides a counterexample to a conjecture on lagrangian surfaces formulated in barja et al 2007 3

singular parabolic equations measures satisfying density conditions and gradient integrability abstract we consider solutions to singular parabolic equations with measurable dependence on the x t variables and having on the right hand side a measure satisfying a density condition we prove that the less the measure is concentrated the more the gradient is regular in the marcinkiewicz scale we provide local estimates and recover some classic results

chern slopes of simply connected complex surfaces of general type are dense in 2 3 we prove that for any number r 2 2 3 there are spin resp nonspin and minimal simply connected complex surfaces of general type x with c 2 x c2 x arbitrarily close to r in particular this shows the existence of simply connected surfaces of general type arbitrarily close to the bogomolov miyaoka yau line in addition we prove that for any r2 1 3 and any integer q 0 there are minimal complex surfaces of general type x with c 2 x c2 x arbitrarily close to r and 1 x isomorphic to the fundamental group of a compact riemann surface of genus q

fibered algebraic surfaces with low slope the background of this paper is the following conjecture on complex surfaces of general type let s be a smooth projective minimal surface of general type over ti with c 2 throughout the paper

borderline gradient continuity for nonlinear parabolic systems we consider the evolutionary p laplacean system begin aligned partial t u triangle p u f qquad p frac 2n n 2 end aligned in cylindrical domains of mathbb r n times mathbb r and prove the continuity of the spatial gradient du under the lorentz space assumption f in l n 2 1 when f is time independent the condition improves in f in l n 1 this is the limiting case of a result of dibenedetto claiming that du is holder continuous when f in l q for q n 2 at the same time this is the natural nonlinear parabolic analog of a linear result of stein claiming the gradient continuity of solutions to the linear elliptic system triangle u in l n 1 is continuous new potential estimates are derived and moreover suitable nonlinear potentials are used to describe fine properties of solutions

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

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

fibred surfaces varieties isogenous to a product and related moduli spaces a fibration of an algebraic surface s over a curve b with fibres of genus at least 2 has constant moduli iff it is birational to the quotient of a product of curves by the action of a finite group g a variety isogenous to a higher product is the quotient of a product of curves of genus at least 2 by the free action of a finite group theorem b gives a characterization of surfaces isogenous to a higher product in terms of the fundamental group and of the euler number theorem c classifies the groups thus occurring and shows that after fixing the group and the euler number one obtains an irreducible moduli space the result of theorem b is extended to higher dimension in theorem g thus generalizing cf also theorem h results of jost yau and mok concerning varieties whose universal cover is a polydisk theorem a shows that fibrations where the fibre genus and the genus of the base b are at least 2 are invariants of the oriented differentiable structure the main theorems d and e characterize surfaces carrying constant moduli fibrations as surfaces having a zariski open set satisfying certain topological conditions e g having the right euler number the right fundamental group and the right fundamental group at infinity 0 introduction the study of fibrations of a smooth algebraic surface s over a smooth algebraic curve b lies at the heart of the classification theory and of the geometry of algebraic surfaces our main results in this paper theo rems d e concern a topological characterization of surfaces admitting a constant moduli fibration these results are intimately related to a series of variations on the same theme also in higher dimension and to a circle of allied techniques which we present in the course of the paper the first new result we have in this direction and which is presented in section 2 after we recall several known results some classical some quite recent is the following theorem on the dif ferentiable invariance of the genus of fibres cf theorem 2 9 for a more precise statement theorem a let f s b be a fibration of the surface s over a curve b of genus b 2 and with fibre f of genus g 1 assume that s is another smooth compact kahler surface which is orientedly diffeomorphic to s then the subspace of h 1 s corresponding to f h 1 b determines a fibration f s b with

steepest descent using smoothed gradients this paper reviews solving differential equations using the least squares steepest descent method with sobolev gradients the method s superiority over standard steepest descent with a euclidean gradient is explained in terms of stability and the classical courant freiderichs lewy condition for the path of steepest descent the spectra for several of the operators arising from a cononical example are also computed kantorovich s inequality then gives explicit estimates on the rate of convergence for the two processes in this way use of the sobolev gradient is viewed as a very effective preconditioning strategy for the linear part of the differential equation

new cycles in the griffiths group of the generic abelian threefold introduction in this paper we give a method to produce infinitely many new 1 cycles homologically but not algebraically equivalent to 0 on the generic abelian threefold the method is based on the construction of the paper some density results for curves with non simple jacobians cf cp in cp we showed in particular that for any g there are infinitely many curves of genus g mapping to a generic principally polarized abelian threefold a the image of each such curve generates a as a group furthermore these curves are rigid up to translation and different in moduli we can use these curves to construct an infinite countable number of 1 cycles on the threefold a which are homologically but not algebraically equivalent to 0 i e elements of gr2 a the griffiths group of a

least squares finite element method and preconditioned conjugate gradient solution a least squares variational procedure for first order systems of differential equations and an approximate formulation based on finite elements are developed error estimates a condition number bound and analysis of weighting factors are given steepest descent and conjugate gradient solution procedures are examined and an appropriate preconditioner constructed which is demonstrated to yield rapid convergence and to be insensitive to problem size numerical studies of rates of convergence for a test problem are given

regularity for parabolic systems of uhlenbeck type with orlicz growth abstract we study the local regularity of p caloric functions or more generally of caloric functions in particular we study local solutions of non linear parabolic systems with homogeneous right hand side where the leading terms have uhlenbeck structure of orlicz type this paper closes the gap of 23 where liebermann proved that if the gradient of a solution is bounded it is holder continuous the crucial step is a novel local estimates for the gradient of the solutions which generalize and improve the pioneering estimates of dibenedetto and friedman 12 10 for the p laplace heat equation

a least squares approach based on a discrete minus one inner product for first order systems the purpose of this paper is to develop and analyze a least squares approximation to a first order system the first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and or nonsymmetric the approach taken here is novel in that the least squares functional employed involves a discrete inner product which is related to the inner product in h 1 the sobolev space of order minus one on the use of this inner product results in a method of approximation which is optimal with respect to the required regularity as well as the order of approximation even when applied to problems with low regularity solutions in addition the discrete system of equations which needs to be solved in order to compute the resulting approximation is easily preconditioned thus providing an efficient method for solving the algebraic equations the preconditioner for this discrete system only requires the construction of preconditioners for standard second order problems a task which is well understood

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

gradient estimates for a class of parabolic systems we establish local calderon zygmund type estimates for a class of parabolic problems whose model is the nonhomogeneous degenerate singular parabolic p laplacian system ut div du p 2du div f p 2f proving that f lqloc du lqloc q p we also treat systems with discontinuous coefficients of vanishing mean oscillation vmo type

a nonlinear parabolic equation with drift term abstract we consider weak solutions to a nonlinear evolution diffusion convection equation with an unbounded convection field we provide existence and uniqueness results as well as the asymptotic behaviour of solutions

xiao s conjecture for general fibred surfaces we prove that the genus g the relative irregularity qf and the clifford index cf of a non isotrivial

riesz potentials and pointwise estimates of solutions to anisotropic porous medium equation abstract for an anisotropic porous medium equation with l 1 regular lower order terms we prove the local boundedness and continuity of weak solutions in terms of the linear riesz potential of the right hand side of equation

on irrationality of surfaces in p3 abstract the degree of irrationality irr x of a n dimensional complex projective variety x is the least degree of a dominant rational map x p n it is a well known fact that given a product x p m or a n dimensional variety y dominating x their degrees of irrationality may be smaller than the degree of irrationality of x in this paper we focus on smooth surfaces s p 3 of degree d 5 and we prove that irr s p m irr s for any integer m 0 whereas irr y irr s occurs for some y dominating s if and only if s contains a rational curve

high order sobolev preconditioning abstract this paper compares the use of first and second order sobolev gradients to solve differential equations using the method of least squares steepest descent the use of high order sobolev gradients offers a very effective preconditioning strategy for the linear part of a nonlinear differential equation

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

holder estimates for nonlinear degenerate parabolic sytems on etablit des estimations de holder pour le gradient d espace u des solutions locales faibles des systemes paraboliques degeneres on etablit la continuite de holder des solutions non negatives d une equation parabolique degeneree

fundamental groups of galois closures of generic projections for the galois closure x gal of a generic projection from a surface x it is believed that 1 x gal gives rise to new invariants of x however in all examples this group is surprisingly simple in this article we offer an explanation for this phenomenon we compute a quotient of 1 x gal that depends on 1 x and data from the generic projection only in all known examples this quotient is in fact isomorphic to 1 x gal as a byproduct we simplify the computations of moishezon teicher and others

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

family of curves with large unitary summand in the hodge bundle in this note we construct non isotrivial families of curves of genus g ge 2 where the rank of the unitary summand contained in the hodge bundle can be as large as 2g 1 3 and hence disprove xiao s conjecture for the unitary rank

marcinkiewicz estimates for degenerate parabolic equations with measure data we consider solutions to degenerate parabolic equations with measurable coefficients having on the right hand side a measure satisfying a suitable density condition we prove integrability results for the gradient in the marcinkiewicz scale

sobolev gradients and the ginzburg landau functional we describe a sobolev gradient method for finding minima of the ginzburg landau functional for superconductivity this method leads to a particularly simple algorithm which avoids consideration of the nonlinear boundary conditions associated with the ginzburg landau equations

the fundamental group of a mathcal cp 2 complement of a branch curve as an extension of a solvable group by a symmetric group the main result in this paper is as follows theorem let s be the branch curve in cp 2 of a generic projection of a veronese surface then 1 cp 2 s is an extension of a solvable group by a symmetric group a group with the property mentioned in the theorem is almost solvable in the sense that it contains a solvable normal subgroup of finite index we pose the following question question for which families of simply connected algebraic surfaces of general type is the fundamental group of the complement of the branch curve of a generic projec tion to cp 2 an extension of a solvable group by a symmetric group

smallness and cancellation in some elliptic systems with measure data abstract in a bounded open subset r n we study dirichlet problems with elliptic systems involving a finite radon measure on r n with values into r n defined by div a x u x d u x in u 0 on where a i x y 1 n j 1 n a i j x y j with 1 n the equation index we prove the existence of a distributional solution u r n obtained as the limit of approximations by assuming i that coefficients a i j are bounded caratheodory functions ii ellipticity of the diagonal coefficients a i j and iii smallness of the quadratic form associated to the off diagonal coefficients a i j i e verifying a r staircase support condition with r 0 such a smallness condition is satisfied for instance in each one of these cases a a i j a j i skew symmetry b a i j is small c a i j may be decomposed into two parts the first enjoying skew symmetry and the second being small in absolute value we give an example that satisfies our hypotheses but does not satisfy assumptions introduced in previous works a brezis s type nonexistence result is also given for general smooth elliptic hyperbolic systems

hodge structures on abelian varieties of type iv let a be a general member of a pel family of abelian varieties with endomorphisms by an imaginary quadratic number field k and let e be an elliptic curve with complex multiplications by k we show that the usual hodge conjecture for products of a with powers of e implies the general hodge conjecture for all powers of a we deduce the general hodge conjecture for all powers of certain 5 dimensional abelian varieties

on the second sectional h arithmetic genus of polarized manifolds let x l be a polarized variety defined over the complex number field with dim x n in this paper we introduce the notion of the i th sectional h arithmetic genus hi x l for every integer i with 0 i n we expect that this invariant has a property similar to the euler poincare characteristic of the structure sheaf of i dimensional varieties in this paper we consider the case where x is smooth and i 2 and we study a polarized version of some results in the theory of surfaces

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

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

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

topologically invariant chern numbers of projective varieties abstract we prove that a rational linear combination of chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the euler and pontryagin numbers in dimension at least three we prove that only multiples of the top chern number which is the euler characteristic are invariant under diffeomorphisms that are not necessarily orientation preserving these results solve a long standing problem of hirzebruch s we also determine the linear combinations of chern numbers that can be bounded in terms of betti numbers

boundary riesz potential estimates for elliptic equations with measurable nonlinearities abstract we consider elliptic equations with measurable nonlinearities in a half cube q 2 r x 1 x r n x 1 0 where the boundary data is given on q 2 r x 1 x r n x 1 0 we obtain a point wise estimate of the gradient in terms of riesz potential of the right hand side data and the oscillation of the gradient of the boundary data under the assumption that the nonlinearity is only allowed to be measurable in x 1 variable

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

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

on the classification of irregular surfaces of general type with nonbirational bicanonical map the present paper is devoted to the classification of irregular surfaces of general type with pg 3 and nonbirational bicanonical map our main result is that if s is such a surface and if s is minimal with no pencil of curves of genus 2 then s is the symmetric product of a curve of genus 3 and therefore pg q 3 and k2 6 furthermore we obtain some results towards the classification of minimal surfaces with pg q 3 such surfaces have 6 kz 9 and we show that kz 6 if and only if s is the symmetric product of a curve of genus 3 we also classify the minimal surfaces with pg q 3 with a pencil of curves of genus 2 proving in particular that for those one has kz 8

families of galois closure curves for plane quintic curves abstract in yoshihara 2003 14 yoshihara studied a family of curves of genus ten over a smooth plane quartic curve which is called a family of galois closure curves and determined the types of singular fibers in this paper we consider similar problem for a smooth quintic plane curve consequently we obtain smooth projective minimal surfaces of general type with c 1 2 2 c 2

rational orbits on three symmetric products of abelian varieties let a be an n dimensional abelian variety n 2 let ch 0 a be the group of zero cycles of a modulo rational equivalence by regarding an effective degree k zero cycle as a point on s k a the k symmetric product of a and by considering the associated rational equivalence class we get a map s k a ch 0 a whose fibres are called orbits for any n 2 in this paper we determine the maximal dimension of the orbits when k 2 or 3 it is respectively 1 and 2 and the maximal dimension of families of orbits moreover for generic a we get some refinements and in particular we show that if dim a 4 s 3 a does not contain any orbit note that it implies that a generic abelian four fold does not contain any trigonal curve

unbounded supersolutions of some quasilinear parabolic equations a dichotomy abstract we study unbounded supersolutions of the evolutionary p laplace equation with slow diffusion they are the same functions as the viscosity supersolutions a fascinating dichotomy prevails either they are locally summable to the power p 1 n p 0 or not summable to the power p 2 there is a void gap between these exponents those summable to the power p 2 induce a radon measure while those of the other kind do not we also sketch similar results for the porous medium equation

quantitative asymptotic estimates for evolution problems abstract we study a quantitative asymptotic stability estimates for solutions to nonlinear evolution equations more precisely we measure the distance in time of a solution to a parabolic problem from a solution to a stationary one