postprocessing the galerkin method the finite element case a postprocessing technique developed earlier for spectral methods is extended here to galerkin finite element methods for dissipative evolution partial differential equations the postprocessing amounts to solving a linear elliptic problem on a finer grid or higher order space once the time integration on the coarser mesh is completed this technique increases the convergence rate of the finite element method to which it is applied and this is done at almost no additional computational cost the numerical experiments presented here show that the resulting postprocessed method is computationally more efficient than the method to which it is applied say quadratic finite elements as well as standard methods of similar order of convergence as the postprocessed one say cubic finite elements the error analysis of the new method is performed in l2 and in l infty norms

multiple positive solutions for a quasilinear elliptic equation involving singular potential and critical sobolev exponent in this paper we are concerned with a class of quasilinear elliptic equation in which the potential is singular at the origin and the nonlinearity is p sublinear near the origin by the ekeland variational principle and mountain pass theorem we prove that there exist at least two positive solutions under suitable assumptions on the potential and the nonlinearity

optimal error bounds for two grid schemes applied to the navier stokes equations abstract we consider two grid mixed finite element schemes for the spatial discretization of the incompressible navier stokes equations a standard mixed finite element method is applied over the coarse grid to approximate the nonlinear navier stokes equations while a linear evolutionary problem is solved over the fine grid the previously computed galerkin approximation to the velocity is used to linearize the convective term for the analysis we take into account the lack of regularity of the solutions of the navier stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data optimal error bounds are obtained

two level method based on finite element and crank nicolson extrapolation for the time dependent navier stokes equations a fully discrete two level finite element method the two level method is presented for solving the two dimensional time dependent navier stokes problem the method requires a crank nicolson extrapolation solution u h tau 0 p h tau 0 on a spatial time coarse grid j h tau 0 and a backward euler solution u h tau p h tau on a space time fine grid j h tau the error estimates of optimal order of the discrete solution for the two level method are derived compared with the standard crank nicolson extrapolation method the one level method based on a space time fine grid j h tau the two level method is of the error estimates of the same order as the one level method in the h1 norm for velocity and the l2 norm for pressure however the two level method involves much less work than the one level method

nontrivial solutions to semilinear elliptic problems involving two critical hardy sobolev exponents abstract in this paper we investigate a semilinear elliptic equation which involves doubly critical hardy sobolev exponents and a hardy type term by means of the linking theorem and delicate energy estimates the existence of nontrivial solutions to the problem is established

the brezis nirenberg problem for the laplacian with a singular drift in rn and sn abstract we consider the brezis nirenberg problem for the laplacian with a singular drift for a geodesic ball in both r n and s n 3 n 5 the singular drift we consider derives from a potential which is symmetric around the center of the geodesic ball here the potential is given by a parameter say times the logarithm of the distance to the center of the ball in both cases we determine the exact region in the parameter space for which positive smooth solutions of this problem exist and the exact region for which there are no solutions the parameter space is characterized by the geodesic radius of the ball and the coupling constant of the linear term of the brezis nirenberg problem

maximum l 2 convergence rates of the crank nicolson scheme to the stokes initial value problem let a denote the stokes operator and d a alpha the domain of its fractional powers a alpha we consider the homogeneous stokes initial value problem with initial data u 0 u 0 in d a 1 varepsilon varepsilon in 0 1 for stokes like equations the range varepsilon in 0 frac 1 4 is of special interest as any solution derived from varepsilon ge frac 1 4 would necessarily have to satisfy an additional in practice unverifiable compatibility condition at time t 0 approximating any strong solution u in c 0 0 infty d a 1 varepsilon in time direction on a finite time interval 0 t with a crank nicolson scheme we show convergence of order o frac tau 2 t 1 varepsilon which is maximal for the assumed regularity of the data and reflects the loss of regularity as t rightarrow 0 the error estimates are derived by energy and semigroup methods combined with a parabolic duality argument

existence of multiple positive weak solutions and estimates for extremal values to a class of elliptic problems with hardy term and singular nonlinearity abstract this paper is devoted to the study of a class of elliptic equations of the form u x 2 u h x u q w x u p x 0 with dirichlet boundary conditions where 0 r n n 3 is a bounded domain with smooth boundary 0 is a parameter 0 n 2 2 4 0 q 1 p 2 1 n 2 n 2 h x 0 and w x is a given function with the set x w x 0 of positive measure using variational methods we establish some existence and multiplicity of positive solutions and provide uniform estimates of extremal values for the problem

a spectral element method for the navier stokes equations with improved accuracy we present a procedure to enhance the accuracy of spectral element methods for the evolutionary navier stokes equations the approximations to the velocity and the pressure obtained using a spectral element method are postprocessed by solving a discrete stokes problem only once when the time evolution is completed the analysis shows that this procedure improves the order of convergence for the approximations to the velocity and the pressure the choice of a discrete pressure space endowed with an inf sup condition independent of the discretization parameter leads to optimal error bounds for the pressure

the eigenvalue problem and infinitely many sign changing solutions for an elliptic equation with critical hardy constant abstract by li yau s idea and a more precise hardy s inequality we obtain a new lower bound of eigenvalues for an elliptic equation with critical hardy constant then using the lower bounds we prove that there are infinitely many sign changing solutions for a class of elliptic equation with critical hardy constant which extends the results in schechter and zou 2005 9 to the critical hardy constant case

two level method and some a priori estimates in unsteady navier stokes calculations a two level method proposed for quasielliptic problems is adapted in this paper to the simulation of unsteady incompressible navier stokes flows the method requires a solution of a nonlinear problem on a coarse grid and a solution of linear symmetric problem on a fine grid the scaling between these two grids is superlinear approximation stability and convergence aspects of a fully discrete scheme are considered stability properties of the two level scheme are compared with those for a commonly used semi implicit scheme some new estimates are also proved for the latter

on the quasilinear elliptic problems with critical sobolev hardy exponents and hardy terms abstract let r n be a smooth bounded domain such that 0 n 3 in this paper we study the critical quasilinear elliptic problems p u u p 2 u x p u p t 2 x t u u q 2 x s u u w 0 1 p with dirichlet boundary condition where p u div u p 2 u 1 p n 0 n p p p 0 0 s t p p q p s p n s n p p t p n t n p p s and p t are the critical sobolev hardy exponents via variational methods we deal with the conditions that ensure the existence of positive solutions for the equation the results depend crucially on the parameters p q s and

bifurcation currents in holomorphic dynamics on p k potential theory has been introduced in one dimensional rational dynamics by brolin and tortrat 4 29 but does not play a central role there in higher dimension however as the classical tools are not any longer efficient pluri potential theory has revealed itself to be essential the fundamental works of hubbard papadopol fornaess sibony bedfordsmillie briend duval see 26 for precise references enlight the remarkable efficency of pluri potential theory in holomorphic dynamics on pk or ck it is therefore tempting to study the parameter spaces in a similar way more precisely one would like to relate the bifurcations of an holomorphic family f x of endomorphisms of pk to the powers of a certain current on the parameter space x let us recall that in dimension k 1 a bifurcation is said to occur at some point 0 x if the julia set of f does not move continuously around 0 the famous work of mane sadsullivan 16 which is based on the lemma and the fatou cremer sullivan classification relates the bifurcations with the instability of the critical orbits it also establishes that the bifurcations concentrate on the complement of an open dense subset of x for the quadratic family z2 x c the bifurcation locus is precisely the boundary of the mandelbrot set

multi level spectral galerkin method for the navier stokes problem i spatial discretization a multi level spectral galerkin method for the two dimensional non stationary navier stokes equations is presented the method proposed here is a multiscale method in which the fully nonlinear navier stokes equations are solved only on a low dimensional space subsequent approximations are generated on a succession of higher dimensional spaces j 2 j by solving a linearized navier stokes problem around the solution on the previous level error estimates depending on the kinematic viscosity 0 1 are also presented for the j level spectral galerkin method the optimal accuracy is achieved when we demonstrate theoretically that the j level spectral galerkin method is much more efficient than the standard one level spectral galerkin method on the highest dimensional space

postprocessing finite element methods for the navier stokes equations the fully discrete case an accuracy enhancing postprocessing technique for finite element discretizations of the navier stokes equations is analyzed the technique had been previously analyzed only for semidiscretizations and fully discrete methods are addressed in the present paper we show that the increased spatial accuracy of the postprocessing procedure is not affected by the errors arising from any convergent time stepping procedure further refined bounds are obtained when the time stepping procedure is either the backward euler method or the two step backward differentiation formula

solutions of the quasilinear elliptic problem with a critical sobolev hardy exponent and a hardy type term abstract in the present paper the quasilinear elliptic problem with a critical sobolev hardy exponent and a hardy type term is considered by means of a variational method the existence of nontrivial solutions to the problem is obtained

on the elliptic problems involving multi singular inverse square potentials and multi critical sobolev hardy exponents abstract let r n n 3 be a smooth bounded domain such that the different points a i i 1 2 k k 2 0 i n 2 2 2 0 s i 2 p i 2 n s i n 2 are the critical sobolev hardy exponents we deal with the conditions that ensure the existence of positive solutions for the multi singular and multi critical elliptic problem u i 1 k i x a i 2 u i 1 k u p i 1 x a i s i with the dirichlet boundary condition involving the hardy inequality and the sobolev hardy inequality the results depend crucially on the parameters s i and i i 1 2 k

multiple positive solutions for a semilinear equation with critical exponent and prescribed singularity abstract this paper is devoted to the study of a semilinear equation of the form u v x u u 2 2 u h x u h 0 1 where r n n 3 is an open bounded domain with smooth boundary 0 n 2 2 2 2 2 n n 2 h x 0 and v x has prescribed finitely many singular points using variational methods we establish some existence and multiplicity of positive solutions for the problem

the postprocessed mixed finite element method for the navier stokes equations a postprocessing technique for mixed finite element methods for the incompressible navier stokes equations is studied the technique was earlier developed for spectral and standard finite element methods for dissipative partial differential equations the postprocessing amounts to solving a stokes problem on a finer grid or higher order space once the time integration on the coarser mesh is completed the analysis presented here shows that this technique increases the convergence rate of both the velocity and the pressure approximations numerical experiments are presented that confirm both this increase in the convergence rate and the corresponding improvement in computational efficiency

nodal and positive solutions for singular semilinear elliptic equations with critical exponents in symmetric domains abstract in this paper we consider the semilinear elliptic problem u u x 2 u k x u 2 2 u in u 0 on where is a smooth bounded domain in r n n 4 0 n 2 2 4 2 2 n n 2 is the critical sobolev exponent k x is a continuous function when and k x are invariant under a group of orthogonal transformations we prove the existence of nodal and positive solutions for 0 1 where 1 is the first dirichlet eigenvalue of x 2 on

the postprocessed mixed finite element method for the navier stokes equations refined error bounds a postprocessing technique for mixed finite element methods for the incompressible navier stokes equations is analyzed the postprocess which amounts to solving a linear stokes problem is shown to increase the order of convergence of the method to which it is applied by one unit times the logarithm of the mesh diameter in proving the error bounds some superconvergence results are also obtained contrary to previous analysis of the postprocessing technique in the present paper we take into account the loss of regularity suffered by the solutions of the navier stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data as in h g heywood and r rannacher siam j numer anal 25 1988 pp 489 512 where the same hypothesis is assumed no better than fifth order convergence is achieved

multiple positive solutions for semilinear elliptic equations involving multi singular inverse square potentials and concave convex nonlinearities abstract in this paper we deal with the existence and multiplicity of positive solutions for the multi singular semilinear elliptic equations u i 1 k i x a i 2 u u 2 2 u u q 2 u x where r n n 3 is a smooth bounded domain such that the points a i i 1 2 k k 2 are different 0 i n 2 2 2 0 1 q 2 and 2 2 n n 2 the results depend crucially on the parameters q and i for i 1 2 k

solutions to critical elliptic equations with multi singular inverse square potentials abstract let be an open bounded domain in r n n 3 with smooth boundary we are concerned with the multi singular critical elliptic problem u i 1 k i u x a i 2 q x u 2 2 u u h 0 1 where i r 2 2 n n 2 a i 1 i k and q x is a positive bounded function on using moser iteration we prove the asymptotic behavior of solutions for at points a i 1 i k by exploiting the effect of the coefficient of the critical nonlinearity we by means of a variational method establish the existence of positive and sign changing solutions for problem

multiple positive solutions for a semilinear elliptic equation with critical sobolev exponent in this paper we study a class of semilinear elliptic equations with hardy potential and critical sobolev exponent by means of the ekeland variational principle and mountain pass theorem multiple positive solutions are obtained

on the existence and multiplicity of solutions for a class of singular elliptic problems abstract in this paper we prove the existence and multiplicity of solutions for a class of elliptic problems with singular coefficients we also prove the existence of a sign changing solution for a brezis nirenberg type problem with hardy potential and singular coefficients our proofs are based on the krasnoselskii genus the concentration compactness principle of lions and the linking theorem

finite element approximation of the nonstationary navier stokes problem part iii smoothing property and higher order error estimates for spatial discretization this paper continues our error analysis of finite element galerkin approximation of the nonstationary navier stokes equations optimal order error estimates both local and global are derived for higher order finite elements under appropriate assumptions about the smoothness and stability of the solution these assumptions take into account the loss of regularity at t 0 that one generally has to expect in the absence of higher order nonlocal compatibility conditions for the data of the problem

existence of solutions for a nonlinear pde with an inverse square potential abstract via linking theorem and delicate energy estimates the existence of nontrivial solutions for a nonlinear pde with an inverse square potential and critical sobolev exponent is proved this result gives a partial positive answer to an open problem proposed in ferrero and gazzola j differential equations 177 2001 494

stability of higher order hood taylor methods the stability of a higher order hood taylor method for the approximation of the stationary stokes equations using continuous piecewise polynomials of degree 3 to approximate velocities and continuo

the postprocessing galerkin and nonlinear galerkin methods a truncation analysis point of view we revisit the postprocessing algorithm and give a justification from a classical truncation analysis point of view we assume a perturbation expansion for the high frequency mode component of solutions to the underlying equation keeping terms to certain orders we then generate approximate systems which correspond to numerical schemes we show that the first two leading order methods are in fact the postprocessed galerkin and postprocessed nonlinear galerkin methods respectively hence postprocessed galerkin is a natural leading order method more natural than the standard galerkin method for approximating solutions of parabolic dissipative pdes the analysis is presented in the framework of the two dimensional navier stokes equation nse however similar analysis may be done for any parabolic dissipative nonlinear pde the truncation analysis is based on asymptotic estimates in time for the low and high mode components we also introduce and investigate an alternative postprocessing scheme whic

on the elliptic problems with critical weighted sobolev hardy exponents abstract let r n be a smooth bounded domain such that 0 n 3 in this paper we deal with the conditions that ensure the existence of nontrivial solutions for the elliptic equation div x 2 a u u x 2 1 a u p 2 x b p u u with dirichlet boundary condition which involving the caffarelli kohn nirenberg inequalities the results depend crucially on the parameters a b and

an exact estimate result for a semilinear equation with critical exponent and prescribed singularity abstract we consider the singular boundary value problem u v x u h x u u 2 1 in u x 0 in u x 0 on where 0 n 2 2 2 0 1 2 2 n n 2 h x is a given function and v x has prescribed finitely many singular points our goal in this paper is to establish some existence and multiplicity results for above problem when 0 for some 0 and obtain exact estimate for extremal value 2 h x 0 for above problem

stability and bifurcations for dissipative polynomial automorphisms of mathbb c 2 we study stability and bifurcations in holomorphic families of polynomial automorphisms of mathbb c 2 we say that such a family is weakly stable over some parameter domain if periodic orbits do not bifurcate there we first show that this defines a meaningful notion of stability which parallels in many ways the classical notion of j stability in one dimensional dynamics define the bifurcation locus to be the complement of the weak stability locus in the second part of the paper we prove that under an assumption of moderate dissipativity the parameters displaying homoclinic tangencies are dense in the bifurcation locus this confirms one of palis conjectures in the complex setting the proof relies on the formalism of semi parabolic bifurcation and the construction of critical points in semi parabolic basins which makes use of the classical denjoy carleman ahlfors and wiman theorems

elliptic problems with critical exponents and hardy potentials this paper is devoted to the existence of positive solutions of a dirichlet problem with critical exponent and a singular potential under various assumption on the domain omega which include some kinds of unbounded domains we prove the existence of ground states and of symmetric solutions c 2002 elsevier science usa all rights reserved

multiplicity result for a singular elliptic equation with indefinite nonlinearity via constraint minimization argument and delicate energy estimates we show the existence of two positive solutions for a singular elliptic equation with indefinite nonlinearity

on the singular critical quasilinear problems in rn abstract in this paper we study the following quasilinear problem p u u p 2 u u p 2 u x p u p 2 u f u u w r 1 p r n where n 3 2 p n p n p n p and f u satisfies some kinds of assumptions by employing the variational methods and analytic techniques we obtain the existence of a nontrivial solution to the problem under some assumptions for the parameters p n and at last we manage to prove that the problem has at least a pair of nodal solutions u k for each k n and both u k possess exactly k nodes

a posteriori error estimations for mixed finite element approximations to the navier stokes equations a posteriori estimates for mixed finite element discretizations of the navier stokes equations are derived we show that the task of estimating the error in the evolutionary navier stokes equations can be reduced to the estimation of the error in a steady stokes problem as a consequence any available procedure to estimate the error in a stokes problem can be used to estimate the error in the nonlinear evolutionary problem a practical procedure to estimate the error based on the so called postprocessed approximation is also considered both the semidiscrete in space and the fully discrete cases are analyzed some numerical experiments are provided

weak eigenfunctions for the linearization of extremal elliptic problems abstract we consider the semilinear elliptic problem formula where is a nonnegative parameter and g is a positive nondecreasing convex nonlinearity there exists a value of the parameter which is extremal in terms of existence of solution we study the linearization of the semilinear problem at the extremal weak solution corresponding to the parameter in some cases this linearized problem has discrete and positive h 1 0 spectrum however we prove that there always exists a positive weak eigenfunction in l 1 with eigenvalue zero for this linearized problem the zero l 1 eigenvalue is coherent with the nonexistence of solutions of the semilinear problem for finally we find all weak eigenfunctions and eigenvalues for the linearization of the extremal problem when is the unit ball and g u e u or g u 1 u p

postprocessing the linear finite element method we extend the idea of the postprocessing galerkin methods for dissipative evolution equations to the case of the linear finite element method the postprocessing technique has been developed earlier for spectral methods and for higher order finite element methods the analysis shows that this procedure improves the order of convergence of the piecewise linear galerkin finite element approximation in the h1 norm we show by means of numerical experiments that there is no improvement in the order of convergence in the l2 norm

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

postprocessing for stochastic parabolic partial differential equations we investigate the strong approximation of stochastic parabolic partial differential equations with additive noise we introduce postprocessing in the context of a standard galerkin approximation although other spatial discretizations are possible in time we follow g j lord and j rougemont ima j numer anal 24 2004 pp 587 604 and use an exponential integrator we prove strong error estimates and discuss the best number of postprocessing terms to take numerically we evaluate the efficiency of the methods and observe rates of convergence some experiments with the implicit euler maruyama method are described

multiple positive solutions for a quasilinear elliptic problem involving critical sobolev hardy exponents and concave convex nonlinearities abstract let r n n 3 be a bounded smooth domain containing the origin in this paper by using variational methods the multiplicity of positive solutions is obtained for a quasilinear elliptic problem p u u p 2 u x p u p t 2 x t u u q 2 x s u u w 0 1 p with dirichlet boundary condition where p u div u p 2 u 1 p n 0 n p p p 0 0 s t p 1 q p and p t p n t n p is the critical sobolev hardy exponent

multiple positive and sign changing solutions of an elliptic equation with fast increasing weight and critical growth abstract we consider the following equation div k x u k x x u q 2 u q x k x u 2 2 u x r n where n 3 2 q 2 2 n n 2 0 is a parameter k x exp x 4 2 2 2 q 2 2 and 0 q x c r n using variational methods and delicate estimates we establish some existence and multiplicity of positive and sign changing solutions for the problem provided that the maximum of q x is achieved at different points

a borderline case in elliptic problems involving weights of caffarelli kohn nirenberg type abstract this work deals with problems div x 2 u n 2 1 2 2 u x 2 1 f x u u 0 in u 0 on with different kinds of nonlinear functions f x u that will be specified in each statement the main difficulties come from the presence of the constant n 2 1 2 2 which is the optimal one in the hardy sobolev inequalities this leads to a nontrivial adaptation of the functional framework

postprocessing the finite element method for semilinear parabolic problems in this paper we consider postprocessing of the finite element method for semilinear parabolic problems the postprocessing amounts to solving a linear elliptic problem on a finer grid or higher order space once the time integration on the coarser mesh is completed the convergence rate is increased at almost no additional computational cost this procedure was introduced and analyzed in garcia archilla and titi siam j numer anal 37 2000 pp 470 499 we extend the analysis to the fully discrete case and prove error estimates for both space and time discretization the analysis is based on error estimates for the approximation of time derivatives by difference quotients

positive solutions for neumann elliptic problems involving critical hardy sobolev exponent with boundary singularities abstract in this paper we consider the existence of positive solution for some semilinear elliptic equations with neumann boundary condition involving a critical hardy sobolev exponent and hardy terms with boundary singularities using mountain pass lemma without ps condition and the strong maximum principle we get the existence of a positive solution

multiple positive and sign changing solutions for a singular schrodinger equation with critical growth abstract variational methods are used to prove the existence of multiple positive and sign changing solutions for a schrodinger equation with singular potential having prescribed finitely many singular points some exact local behavior for positive solutions obtained here are also given the interesting aspects are two one is that one singular point of the potential v x and one positive solution can produce one sign changing solution of the problem the other is that each sign changing solution changes its sign exactly once

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

a note on the sign changing solutions to elliptic problems with critical sobolev and hardy terms let rn be a bounded domain such that 0 n 7 2 2nn 2 we obtain existence of sign changing solutions for the dirichlet problem u u x 2 u 2 2u u on u 0 on for suitable positive numbers and

existence of positive solutions for a class of degenerate or singular equations abstract in this paper we consider the class of equations div x 2 a u a x 2 a u u x b p x r n 0 where a b r n 2 a r 0 and 0 p 2 n n 2 1 a b in various range of the parameters involved we obtain both results of existence and nonexistence of positive solutions by combining the mountain pass lemma and the pohozaev type identity

solutions for a singular critical growth problem with a weight in this paper we consider some semilinear elliptic equations with hardy potential by using linking theorem in p rabinowitz minimax methods in critical points theory with applications to differential equations cbms reg conf ser math vol 65 amer math soc providence ri 1986 and analyzing the effect of nonlinearities we establish the existence of nontrivial solutions

existence of multiple positive solutions to singular elliptic systems involving critical exponents abstract this paper is devoted to investigate multiple positive solutions to a singular elliptic system where the nonlinearity involves a combination of concave and convex terms by exploiting the effect of the coefficient of the critical nonlinearity and a variational method we establish the main result which is based on the argument of the compactness

on the quasilinear elliptic problem with a critical hardy sobolev exponent and a hardy term abstract in the present paper a quasilinear elliptic problem with a critical sobolev exponent and a hardy type term is considered by means of a variational method the existence of nontrivial solutions for the problem is obtained the result depends crucially on the parameters p t s and

dynamics of rational maps lyapunov exponents bifurcations and capacity let l f log df d f denote the lyapunov exponent of a rational map f p 1 p 1 in this paper we show that for any holomorphic family of rational maps f x of degree d 1 t f dd c l f defines a natural positive 1 1 current on x supported exactly on the bifurcation locus of the family the proof is based on the following potential theoretic formula for the lyapunov exponent here f c 2 c 2 is a homogeneous polynomial lift of f g f is the escape rate function of f and capk f is the homogeneous capacity of the filled julia set of f we show in particular that the capacity of k f is given explicitly by the formula where res f is the resultant of the polynomial coordinate functions of f we introduce the homogeneous capacity of compact circled and pseudoconvex sets k c 2 and show that the levi measure determined by the geometry of k is the unique equilibrium measure such k c 2 correspond to metrics of non negative curvature on p 1 and we obtain a variational characterization of curvature

on the elliptic problems involving multi singular inverse square potentials and critical sobolev exponent abstract assume that r n n 3 is a smooth bounded domain a i are different points in i 1 2 k k 2 0 2 q 2 n 2 2 2 2 2 n n 2 is the critical sobolev exponent we study the existence and asymptotic behavior of solutions to the multi singular critical elliptic problem u u i 1 k x a i 2 u 2 2 u u q 2 u with dirichlet boundary condition the results depend crucially on the parameters and q

hardy sobolev maz ya type equations in bounded domains abstract we study the regularity palais smale characterization and existence nonexistence of solutions of the hardy sobolev maz ya equation u u y 2 u p t 1 u y t in a bounded domain in r n where x r n is denoted as x y z r k r n k and p t n 2 2 t n 2 we show different behaviors of ps sequences depending on t 0 or t 0

solutions for semilinear elliptic equations with critical exponents and hardy potential abstract in this paper we answer affirmatively an open problem cf theorem 4 in ferrero and gazzola j differential equations 177 2001 494 let 0 be an open bounded domain r n n 5 and assume that 0 n 2 2 2 n 2 n 2 then for all 0 there exists a nontrivial solution with critical level in the range 0 1 n s n 2 for the problem u u x 2 u u 2 2 u in u 0 on

a postprocessed galerkin method with chebyshev or legendre polynomials we present an approximate inertial manifold based postprocess to enhance chebyshev or legendre spectral galerkin methods we prove that the postprocess improves the order of convergence of the galerkin solution yielding the same accuracy as the nonlinear galerkin method numerical experiments show that the new method is computationally more efficient than galerkin and nonlinear galerkin methods new approximation results for chebyshev polynomials are presented

stabilization of galerkin finite element approximations to transient convection diffusion problems a postprocessing technique to improve galerkin finite element approximations to linear evolutionary convection reaction diffusion equations is considered a steady convection reaction diffusion problem with data based on the computed standard galerkin approximation is solved at any fixed time the postprocessing approximation is obtained using the supg method over the same galerkin finite element space error bounds for the method are obtained in the convection dominated regime the numerical experiments we present show a substantial reduction of spurious oscillations achieved by means of this procedure

a small eddy correction method for nonlinear dissipative evolutionary equations considering the interaction between the large and small eddy components of solutions and using the idea of the newton iteration a small eddy correction method is proposed for approximating and numerically solving nonlinear dissipative pdes of parabolic type in particular the navier stokes equations nse we assume that the large eddy approximation to the solution is known formally applying the newton iterative procedure to the small eddy equation we then generate approximate systems it is shown that the first two steps in fact lead to the standard galerkin method sgm and the so called optimum nonlinear galerkin method ong and therefore the small eddy correction method is actually a certain generalization of sgm and ong the boundedness and convergence analysis are presented in the framework of the two dimensional nse the results show that the small eddy correction method can greatly improve the accuracy of sgm approximate solutions

generalized postprocessed approximations to the navier stokes equations based on two grids abstract in this paper we generalize previous postprocessed approximations to the navier stokes equations the idea of postprocessing in the context of mixed finite element approximations is the following once a standard galerkin mixed finite element approximation is computed using a mesh we will call coarse of size h at a fixed time t we postprocess the approximation using a finer mesh of size h h by solving a steady problem with data based on the galerkin approximation in the literature one can find different ways to postprocess the galerkin approximation depending on the problem that is solved over the fine mesh at the final time t basically one can solve a stokes problem an oseen type problem or a newton type problem in the present paper we present a method based on several parameters that generalize the above postprocessing techniques depending on the values chosen for the different parameters one can recover one of the old known postprocessing procedures but also produce a new different method we get error bounds for the generalized method valid for any of the values of the different parameters in all the cases the postprocessed method has a rate of convergence one unit bigger than the rate of convergence of the plain galerkin method in terms of the coarse mesh h and optimal in terms of the fine mesh h the computational added cost of postprocessing is however negligible for the error analysis we do not assume nonlocal compatibility conditions for the true solutions of the navier stokes equations some numerical experiments show the performance of the method for different values of the parameters

solutions of elliptic equations with critical sobolev exponent in dimension three theorem 1 for every a 72 4 there exists a solution of i the previously known results concerning this problem are the following 1 if 1 5 0 i has no solution this follows from pohozaev s identity see lo and from the principle of unique continuation when a 0 we will suppose in the following that i 0 2 every positive solution of i is radial see gidas et al 9 3 let a be the first eigenvalue of a in b with dirichlet boundary condition and p1 the associated eigenfunction then for every a 2 i i has no positive solution this result follows from multiplying i by cp 4 for every a e n2 4 rr there exists a positive solution of i see brezis and nirenberg 5 5 for i 5 n2 4 i has no radial solution see 5 6 for every k e n if 1 e zk with zk k t n k2n2 i has a radial solution with k 1 zeros see atkinson and peletier 2 31 7 the analogue of theorem 1 in dimension n 2 4 holds for general domains see capozzi et al 6 and ambrosetti and struwe l our method is very different and we do not know whether theorem 1 is valid for general domains in ir3

a posteriori error estimates for fully discrete nonlinear parabolic problems abstract we present estimates for the spatial error in fully discrete approximations to nonlinear parabolic problems that extend the a posteriori estimates for the continuous time semi discretization introduced in de frutos and novo j de frutos j novo a posteriori error estimation with the p version of the finite element method for nonlinear parabolic differential equations comput methods appl mech engrg 191 2002 4893 4904 the error indicator is defined as the difference between the fully discrete galerkin and the fully discrete postprocessed approximations we prove that the main term of the temporal error in both the fully discrete galerkin and the postprocessed approximation is the same as a consequence the spatial approximation error can be independently estimated with regard to the time approximation error some numerical experiments are provided

four solutions of an inhomogeneous elliptic equation with critical exponent and singular term abstract in this paper we prove the existence of four nontrivial solutions of u x 2 u u 2 2 u x 2 u f x x 0 and show that at least one of them is sign changing our results extend some previous works such as g tarantello multiplicity results for an inhomogeneous neumann problem with critical exponent manuscripta math 81 1993 51 78 d kang and y deng multiple solutions for inhomogeneous elliptic problems involving critical sobolev hardy exponents j math anal appl 60 2005 729 753 n hirano and n shioji a multiplicty result including a sign changing solution for an inhomogeneous neumann problem with critical exponent proc roy soc edinburgh 137a 2007 333 347

postprocessing the galerkin method a novel approach to approximate inertial manifolds a postprocess of the standard galerkin method for the discretization of dissipative equations is presented the postprocessed galerkin method uses the same approximate inertial manifold phi app to approximate the high wave number modes of the solution as in the nonlinear galerkin nlg method however in this postprocessed galerkin method the value of phi app is calculated only once after the time integration of the standard galerkin method is completed contrary to the nlg in which phi app evolves with time and affects the time evolution of the lower wave number modes the postprocessed galerkin method which is much cheaper to implement computationally than the nlg is shown in the case of fourier modes to possess the same rate of convergence accuracy as the simplest version of the nlg which is based on either the foias manley temam approximate inertial manifold or the euler galerkin approximate inertial manifold this is proved for some problems in one and two spatial dimensions including the navier stokes equations under periodic boundary conditions the advantages of postprocessing that we present here apply not only to the standard galerkin method but also to the computationally more efficient pseudospectral method

existence and multiplicity of positive solutions for some hardy sobolev critical elliptic equation with boundary singularities abstract in this paper by ekeland s variational principle and strong maximum principle we consider the existence and multiplicity of positive solutions for some semilinear elliptic equation involving critical hardy sobolev exponents and hardy terms with boundary singularities

stability and convergence of the crank nicolson adams bashforth scheme for the time dependent navier stokes equations in this paper we study the stability and convergence of the crank nicolson adams bashforth scheme for the two dimensional nonstationary navier stokes equations a finite element method is applied for the spatial approximation of the velocity and pressure the time discretization is based on the crank nicolson scheme for the linear term and the explicit adams bashforth scheme for the nonlinear term moreover we present optimal error estimates and prove that the scheme is almost unconditionally stable and convergent i e stable and convergent when the time step is less than or equal to a constant

the effect of domain topology on the number of positive solutions for singular elliptic problems involving the caffarelli kohn nirenberg inequalities abstract in this paper we consider the critical singular equation involving the caffarelli kohn nirenberg inequalities of the type p 1 div x 2 a u u x 2 1 a u q 2 u x d d u p 2 u x b p x u 0 x u 0 x here is a bounded domain with smooth boundary in r n and contains 0 in its interior 0 a 2 n 2 2 2 n 3 a b a 1 a d a 1 p p a b 2 n n 2 1 a b d d a d 2 n n 2 1 a d is a positive parameter and 2 q d by lusternik schnirelmann category theory we prove that the problem p 1 has at least cat positive solutions

the existence of solutions for quasilinear elliptic problems with combined critical sobolev hardy terms abstract in this paper we investigate the quasilinear elliptic equations with dirichlet boundary conditions and combined critical sobolev hardy terms on bounded smooth domains r n n 3 and prove the existence and multiplicity of solutions by employing ekeland s variational principle

multiple positive solutions for a semilinear equation with prescribed singularity variational methods are used to prove the existence of multiple positive solutions for a semilinear equation with prescribed finitely many singular points some exact local behavior for positive solutions are also given

multiple positive solutions for semilinear elliptic equations with critical weighted hardy sobolev exponents abstract the existence and multiplicity of positive solutions are obtained for a class of semilinear elliptic equations with critical weighted hardy sobolev exponents and the concave convex nonlinearity by variational methods and some analysis techniques

misiurewicz parameters and dynamical stability of polynomial like maps of large topological degree given a family of polynomial like maps of large topological degree we relate the presence of misiurewicz parameters to a growth condition for the volume of the iterates of the critical set this generalizes to higher dimensions the well known equivalence between stability and normality of the critical orbits in dimension one we also introduce a notion of holomorphic motion of asymptotically all repelling cycles and prove its equivalence with other notions of stability our results allow us to generalize the theory of stability and bifurcation developed by berteloot dupont and the author for the family of all endomorphisms of mathbb p k of a given degree to any arbitrary family of endomorphisms of mathbb p k or polynomial like maps of large topological degree

existence and non existence results for quasilinear kirchhoff problems with the hardy sobolev exponent abstract in this paper we are concerned with the following quasilinear elliptic problems of kirchhoff type a b r n u p u p x p d x 1 p u u p 2 u x p u p 2 u x f x u q 2 u x where a 0 b 0 1 0 n p p p 0 p and q 1 p are constants and p u div u p 2 u with 1 p n p p n n p is the hardy sobolev exponent we establish the non existence and the existence of infinitely many nontrivial solutions when 0 for the above problem also for suitable weight function f x we prove the existence of two positive solutions by using nehari manifold and fibering map when 0 is sufficiently small

existence and multiplicity of positive solutions for a class of semilinear elliptic equations involving hardy term and hardy sobolev critical exponents abstract existence and multiplicity of positive solutions are studied for a class of semilinear elliptic equations with hardy term hardy sobolev critical exponents and sublinear nonlinearity by variational methods and some analysis techniques

a posteriori error estimation with the p version of the finite element method for nonlinear parabolic differential equations we analyze an a posteriori error estimator for nonlinear parabolic differential equations in several space dimensions the spatial discretization is carried out using the p version of the finite element method the error estimates are obtained by solving an elliptic problem at the desired times when the estimation is wanted some numerical experiments prove the efficiency of the error estimation