an energy stable fourth order finite difference scheme for the cahn hilliard equation abstract in this paper we propose and analyze an energy stable numerical scheme for the cahn hilliard equation with second order accuracy in time and the fourth order finite difference approximation in space in particular the truncation error for the long stencil fourth order finite difference approximation over a uniform numerical grid with a periodic boundary condition is analyzed via the help of discrete fourier analysis instead of the standard taylor expansion this in turn results in a reduced regularity requirement for the test function in the temporal approximation we apply a second order bdf stencil combined with a second order extrapolation formula applied to the concave diffusion term as well as a second order artificial douglas dupont regularization term for the sake of energy stability as a result the unique solvability energy stability are established for the proposed numerical scheme and an optimal rate convergence analysis is derived in the l 0 t l 2 l 2 0 t h h 2 norm a few numerical experiments are presented which confirm the robustness and accuracy of the proposed scheme

the jacobian determinant revisited this paper is devoted to the study of the jacobian determinant of a map g from a smooth bounded open subset of rn into rn n 2 more generally could be a smooth bounded open subset of an n dimensional manifold starting with the seminal work of c b morrey 27 y reshetnyak 32 and j ball 1 it has been known that one can define the distributional jacobian determinant det g under fairly weak assumption on g in particular it is defined for all maps g w 1 n2 n 1 and also for all

polyharmonic k hessian equations in rn abstract this work is focused on the study of the nonlinear elliptic higher order equation m u s k u f x r n where the k hessian s k u is the kth elementary symmetric polynomial of eigenvalues of the hessian matrix of the solution and the datum f belongs to a suitable functional space this problem is posed in r n and we prove the existence of at least one solution by means of topological fixed point methods for suitable values of m n questions related to the regularity of the solutions and extensions of these results to the nonlocal setting are also addressed on the way to construct these proofs some technical results such as a fixed point theorem and a refinement of the critical sobolev embedding which could be of independent interest are introduced

weak continuity and lower semicontinuity results for determinants weak continuity properties of minors and lower semicontinuity properties of functionals with polyconvex integrands are addressed in this paper in particular it is shown that if un is bounded in open image in new window and if u bv n are such that un u in l1 n and

on the optimality of certain sobolev exponents for the weak continuity of determinants keywords optimality of sobolev exponents weak continuity of determinant radon measure reference caa article 1992 005doi 10 1016 0022 1236 92 90071 pview record in web of science record created on 2008 11 25 modified on 2017 05 12

w1 p quasiconvexity and variational problems for multiple integrals abstract variational problems for the multiple integral i u g u x dx where r m and u r n are studied a new condition on g called w1 p quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of c b morrey it being shown in particular to be necessary for sequential weak lower semicontinuity of i in w 1 p r n and for the existence of minimizers for certain related integrals counterexamples are given concerning the weak continuity properties of jacobians in w 1 p r n p n m an existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses

a new div curl result applications to the homogenization of elliptic systems and to the weak continuity of the jacobian in this paper a new div curl result is established in an open set omega of mathbb r n n geq 2 for the product of two sequences of vector valued functions which are bounded respectively in l p omega n and l q omega n with 1 p 1 q 1 1 n 1 and whose respectively divergence and curl are compact in suitable spaces we also assume that the product converges weakly in w 1 1 omega the key ingredient of the proof is a compactness result for bounded sequences in w 1 q omega based on the imbedding of w 1 q s n 1 into l p s n 1 s n 1 the unit sphere of mathbb r n through a suitable selection of annuli on which the gradients are not too high in the spirit of de giorgi and manfredi the div curl result is applied to the homogenization of equi coercive systems whose coefficients are equi bounded in l rho omega for some rho n 1 over 2 if n 2 or in l 1 omega if n 2 it also allows us to prove a weak continuity result for the jacobian for bounded sequences in w 1 n 1 omega satisfying an alternative assumption to the l infty strong estimate of brezis and nguyen two examples show the sharpness of the results

long time existence for the boussinesq full dispersion systems abstract we establish the long time existence of solutions for the boussinesq full dispersion systems modeling the propagation of internal waves in a two layer system for the two dimensional hamiltonian case b d 0 a 0 c 0 we study the global existence of small solutions of the corresponding system

convergence analysis and error estimates for a second order accurate finite element method for the cahn hilliard navier stokes system in this paper we present a novel second order in time mixed finite element scheme for the cahn hilliard navier stokes equations with matched densities the scheme combines a standard second order crank nicolson method for the navier stokes equations and a modification to the crank nicolson method for the cahn hilliard equation in particular a second order adams bashforth extrapolation and a trapezoidal rule are included to help preserve the energy stability natural to the cahn hilliard equation we show that our scheme is unconditionally energy stable with respect to a modification of the continuous free energy of the pde system specifically the discrete phase variable is shown to be bounded in ell infty left 0 t l infty right and the discrete chemical potential bounded in ell infty left 0 t l 2 right for any time and space step sizes in two and three dimensions and for any finite final time t we subsequently prove that these variables along with the fluid velocity converge with optimal rates in the appropriate energy norms in both two and three dimensions

error estimates for time discretizations of cahn hilliard and allen cahn phase field models for two phase incompressible flows we carry out rigorous error analysis for some energy stable time discretization schemes developed in shen and yang siam j sci comput 32 3 1159 1179 2010 for a cahn hilliard phase field model and in shen and yang chin ann math ser b 31 5 743 758 2010 for an allen cahn phase field model

the distributional hyper jacobian determinants in fractional sobolev spaces abstract in this paper we give a positive answer to the question about hyper jacobian determinants and associated minors raised by baer jerison inspired by recent works of brezis nguyen and baer jerison on jacobian and hessian determinants we establish the weak continuity and fundamental representation for the distributional mth jacobian minors of degree r in the fractional sobolev space w m m r r r n moreover the result is optimal in the framework of fractional sobolev spaces i e all mth jacobian minors of degree r are well defined in w s p r n if and only if w s p r n w m m r m r n

optimal rate convergence analysis of a second order scheme for a thin film model with slope selection abstract in this paper an energy stable second order mixed finite element scheme is proposed and analyzed for the thin film epitaxial growth model with slope selection we employ second order backward differentiation formula bdf scheme with a second order stabilized term which guarantees the long time energy stability to approximate the continuous model in terms of the convergence analysis the key difficulty to derive an optimal rate spatial estimate is associated with the appearance of the gradient operator in the nonlinear terms which may lead to a loss of optimal accuracy order to overcome this well known difficulty we make use of some auxiliary techniques over triangular elements and obtain an optimal convergence rate o h q 1 t 2 in comparison with o h q t 2 rate from a standard projection estimate furthermore we use an efficient preconditioned steepest descent psd solver for the numerical implementation a few numerical examples are presented to validate the stability and convergence

analysis of a mixed finite element method for a cahn hilliard darcy stokes system in this paper we devise and analyze a mixed finite element method for a modified cahn hilliard equation coupled with a nonsteady darcy stokes flow that models phase separation and coupled fluid flow in immiscible binary fluids and diblock copolymer melts the time discretization is based on a convex splitting of the energy of the equation we prove that our scheme is unconditionally energy stable with respect to a spatially discrete analogue of the continuous free energy of the system and unconditionally uniquely solvable we prove that the discrete phase variable is bounded in l infty left 0 t l infty right and the discrete chemical potential is bounded in l infty left 0 t l 2 right for any time and space step sizes in two and three dimensions and for any finite final time t we subsequently prove that these variables converge with optimal rates in the appropriate energy norms in both two and three dimensions

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

on the distributional jacobian of maps from sn into sn in fractional sobolev and holder spaces h brezis and l nirenberg proved that if gk c 0 s n s n and g2 c 0 s n s n n 1 are such that gk g in bmo s n then deggk degg on the other hand if g 2 c 1 s n s n then kronecker s formula asserts that degg 1 jsnj r sn det rg d consequently r sn det rgk d converges to r sn det rg d provided gk g in bmo s n in the same spirit we consider the quantity j g r sn det rg d for all 2 c 1 s n r and study the convergence of j gk in particular we prove that j gk converges to j g for any 2 c 1 s n r if gk converges to g in c 0 s n for some n 1 n surprisingly this result is optimal when n 1 in the case n 1 we prove that if gk g almost everywhere and lim supk 1jgk gjbmo is suciently small then j gk j g for any 2 c 1 s 1 r we also establish bounds for j g which are motivated by the works of j bourgain h brezis and h m nguyen and h m nguyen we pay special attention to the case n 1

a deposition model coupling stokes and darcy s equations with nonlinear deposition abstract in this work we investigate a filtration process whereby particulate is deposited in the flow domain causing the porosity of the region to decrease the fluid flow is modeled as a coupled stokes darcy flow problem and the deposition in the darcy domain is modeled using a nonlinear equation for the porosity existence and uniqueness of a solution to the governing equations is established additionally the nonnegativity and boundedness of the porosity is shown a finite element approximation scheme that preserves the nonnegativity and boundedness of the porosity is investigated accompanying numerical experiments support the analytical findings

optimal function spaces for continuity of the hessian determinant as a distribution abstract we establish optimal continuity results for the action of the hessian determinant on spaces of besov type into the space of distributions on r n in particular inspired by recent work of brezis and nguyen on the distributional jacobian determinant we show that the action is continuous on the besov space of fractional order b 2 2 n n and that all continuity results in this scale of besov spaces are consequences of this result a key ingredient in the argument is the characterization of b 2 2 n n as the space of traces of functions in the sobolev space w 2 n r n 2 on the subspace r n of codimension 2 the most delicate and elaborate part of the analysis is the construction of a counterexample to continuity in b 2 2 n p with p n

a multilevel decoupling method for the navier stokes darcy model abstract this paper considers a multilevel decoupling method for the coupled navier stokes darcy model describing a free flowing fluid over a porous medium the method utilizes a sequence of meshes on which a low dimensional fully coupled nonlinear problem is solved only on a very coarse initial mesh on subsequent finer meshes the approximate solution in each flow region is obtained by solving a linear decoupled problem and performing a correction step the correction step in each domain is achieved by solving a linear system that differs from the original decoupled system only in the right hand side we prove optimal error estimates and demonstrate that for a sequence of meshes with spacing h j h j 1 2 the decoupling method is computationally efficient and achieves the same order of approximation as the fully coupled method

on polyharmonic regularizations of k hessian equations variational methods abstract this work is devoted to the study of the boundary value problem 1 u 1 k s k u f x r n u n u n 2 u n 1 u 0 x where the k hessian s k u is the k th elementary symmetric polynomial of eigenvalues of the hessian matrix and the datum f obeys suitable summability properties we prove the existence of at least two solutions of which at least one is isolated strictly by means of variational methods we look for the optimal values of n that allow the construction of such an existence and multiplicity theory and also investigate how a weaker definition of the nonlinearity permits improving these results