mappings of least dirichlet energy and their hopf differentials the paper is concerned with mappings h colon mathbb x begin array ll rm onto longrightarrow end array mathbb y between planar domains having least dirichlet energy the existence and uniqueness up to a conformal change of variables in mathbb x of the energy minimal mappings is established within the class overline fancyscript h 2 mathbb x mathbb y of strong limits of homeomorphisms in the sobolev space fancyscript w 1 2 mathbb x mathbb y a result of considerable interest in the mathematical models of nonlinear elasticity the inner variation of the independent variable in mathbb x leads to the hopf differential h z overline h bar z rm d z otimes rm d z and its trajectories for a pair of doubly connected domains in which mathbb x has finite conformal modulus we establish the following principle a mapping h in overline fancyscript h 2 mathbb x mathbb y is energy minimal if and only if its hopf differential is analytic in mathbb x and real along partial mathbb x in general the energy minimal mappings may not be injective in which case one observes the occurrence of slits in mathbb x cognate with cracks slits are triggered by points of concavity of mathbb y they originate from partial mathbb x and advance along vertical trajectories of the hopf differential toward mathbb x where they eventually terminate so no crosscuts are created

nucleation barriers for the cubic to tetragonal phase transformation we are interested in the phase transformation from austenite to martensite this transformation is typically accompanied by the generation and growth of small inclusions of martensite we consider a model from geometrically linear elasticity with sharp energy penalization for phase boundaries focusing on a cubicto tetragonal phase transformation we show that the minimal energy for an inclusion of martensite scales like maxfv 2 3 v 9 11 g in terms of the volume v moreover our arguments illustrate the importance of self accommodation for achieving the minimal scaling of the energy the analysis is based on fourier representation of the elastic energy 2012 wiley periodicals inc

on rank one convex functions that are homogeneous of degree one we show that positively 1 homogeneous rank one convex functions are convex at 0 and at matrices of rank one the result is a special case of an abstract convexity result that we establish for positively 1 homogeneous directionally convex functions defined on an open convex cone in a finite dimensional vector space from these results we derive a number of consequences including various generalizations of the ornstein l1 non inequalities most of the results were announced in c r acad sci paris ser i 349 407 409 2011

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

asymptotics in a family of linked strip maps abstract we apply round off to planar rotations obtaining a one parameter family of invertible maps of a two dimensional lattice as the angle of rotation approaches 2 the fourth iterate of the map produces piecewise rectilinear motion which develops along the sides of convex polygons we characterise the dynamics which resembles outer billiards of polygons as the concatenation of so called strip maps each providing an elementary perturbation of an underlying integrable system significantly there are orbits which are subject to an arbitrarily large number of these perturbations during a single revolution resulting in the appearance of a novel discrete space version of near integrable hamiltonian dynamics we study the asymptotic regime of the limiting integrable system analytically and numerically some features of its very rich near integrable dynamics we unveil a dichotomy there is one regime in which the nonlinearity tends to zero and a second where it does not in the latter case numerical experiments suggest that the distribution of the periods of orbits is consistent with that of random dynamics in the former case the fluctuations result in an intricate structure of resonances

existence and uniqueness of global weak solutions to a cahn hilliard stokes darcy system for two phase incompressible flows in karstic geometry abstract we study the well posedness of a coupled cahn hilliard stokes darcy system which is a diffuse interface model for essentially immiscible two phase incompressible flows with matched density in a karstic geometry existence of finite energy weak solution that is global in time is established in both 2d and 3d weak strong uniqueness property of the weak solutions is provided as well

on the gradient set of lipschitz maps we prove that the essential range of the gradient of planar lipschitz maps has a connected rank one convex hull as a corollary in combination with the results in 7 we obtain a complete characterization of incompatible sets of gradients for planar maps in terms of rank one convexity 2008 walter de gruyter berlin new york

uniquely solvable and energy stable decoupled numerical schemes for the cahn hilliard stokes darcy system for two phase flows in karstic geometry we propose and analyze two novel decoupled numerical schemes for solving the cahn hilliard stokes darcy chsd model for two phase flows in karstic geometry in the first numerical scheme we explore a fractional step method operator splitting to decouple the phase field cahn hilliard equation from the velocity field stokes darcy fluid equations to further decouple the stokes darcy system we introduce a first order pressure stabilization term in the darcy solver in the second numerical scheme so that the stokes system is decoupled from the darcy system and hence the chsd system can be solved in a fully decoupled manner we show that both decoupled numerical schemes are uniquely solvable energy stable and mass conservative ample numerical results are presented to demonstrate the accuracy and efficiency of our schemes

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

a weakly nonlinear energy stable scheme for the strongly anisotropic cahn hilliard equation and its convergence analysis abstract in this paper we propose and analyze a weakly nonlinear energy stable numerical scheme for the strongly anisotropic cahn hilliard model in particular a highly nonlinear and singular anisotropic surface energy makes the pde system very challenging at both the analytical and numerical levels to overcome this well known difficulty we perform a convexity analysis on the anisotropic interfacial energy and a careful estimate reveals that all its second order functional derivatives stay uniformly bounded by a global constant this subtle fact enables one to derive an energy stable numerical scheme moreover a linear approximation becomes available for the surface energy part and a detailed estimate demonstrates the corresponding energy stability its combination with an appropriate treatment for the nonlinear double well potential terms leads to a weakly nonlinear energy stable scheme for the whole system in particular such an energy stability is in terms of the interfacial energy with respect to the original phase variable and no auxiliary variable needs to be introduced this has important implications for example in the case that the method needs to satisfy a maximum principle more importantly with a careful application of the global bound for the second order functional derivatives an optimal rate convergence analysis becomes available for the proposed numerical scheme which is the first such result in this area meanwhile for a cahn hilliard system with a sufficiently large degree of anisotropy a willmore or biharmonic regularization has to be introduced to make the equation well posed for such a physical model all the presented analyses are still available the unique solvability energy stability and convergence estimate can be derived in an appropriate manner in addition the fourier pseudo spectral spatial approximation is applied and all the theoretical results could be extended for the fully discrete scheme finally a few numerical results are presented which confirm the robustness and accuracy of the proposed scheme

a robust solver for a second order mixed finite element method for the cahn hilliard equation abstract we develop a robust solver for a second order mixed finite element splitting scheme for the cahn hilliard equation this work is an extension of our previous work in which we developed a robust solver for a first order mixed finite element splitting scheme for the cahn hilliard equation the key ingredient of the solver is a preconditioned minimal residual algorithm with a multigrid preconditioner whose performance is independent of the spatial mesh size and the time step size for a given interfacial width parameter the dependence on the interfacial width parameter is also mild

distribution of periodic orbits for the casati prosen map on rational lattices abstract we study numerically the periodic orbits of the casati prosen map a two parameter reversible map of the torus with zero entropy for rational parameter values this map preserves rational lattices and each lattice decomposes into periodic orbits we consider the distribution function of the periods over prime lattices and its dependence on the parameters of the map based on extensive numerical evidence we conjecture that asymptotically almost all orbits are symmetric and that for a set of rational parameters having full density the distribution function approaches the gamma distribution r x 1 e x 1 x these properties which have been proved to hold for random reversible maps were previously thought to require a stronger form of deterministic randomness such as that displayed by rational automorphisms over finite fields furthermore we show that the gamma distribution is the limit of a sequence of singular distributions which are observed on certain lines in parameter space our experiments reveal that the convergence rate to r is highly non uniform in parameter space being slowest in sharply defined regions reminiscent of resonant zones in hamiltonian perturbation theory

tartar s conjecture and localization of the quasiconvex hull in mathbb r 2 times 2 we give a concrete and surprisingly simple characterization of compact sets k subset mathbb r 2 times 2 for which families of approximate solutions to the inclusion problem du k are compact in particular our condition is algebraic and can be tested algorithmically we also prove that the quasiconvex hull of compact sets of 2 2 matrices can be localized this is false for compact sets in higher dimensions in general

finite dimensional global attractor of the cahn hilliard brinkman system abstract our aim in this paper is to study the asymptotic behavior of the cahn hilliard brinkman system we obtain the existence of a finite fractal dimensional global attractor a in h 4 v i by asymptotical a priori estimates for the cahn hilliard brinkman system

quasiregular mappings and young measures w 1 p gradient young measures supported in the set q 2 k of two dimensional k quasiconformal matrices are studied we prove that these young measures can be generated by gradients of k quasiregular mappings this leads for example to the 0 1 law for quasiregular w 1 p gradient young measures and other quasiregular properties such as higher integrability

optimal distributed control of the cahn hilliard brinkman system with regular potential abstract the main objective of this paper is to study the optimal distributed control of the three dimensional cahn hilliard brinkman system with regular potential we apply recently proved well posedness and regularity results in order to establish the existence of optimal controls as well as the first order necessary optimality conditions for an associated optimal control problem in which a distributed control is applied to the fluid flow

a novel second order linear scheme for the cahn hilliard navier stokes equations abstract in this paper we consider the cahn hilliard equation coupled with the incompressible navier stokes equation usually known as the cahn hilliard navier stokes chns system the chns system has been widely embraced to investigate the dynamics of a binary fluid mixture by utilizing the modified leap frog time marching method we propose a novel numerical algorithm for solving the chns system in an efficient and accurate manner this newly proposed scheme has several advantages first of all the proposed scheme is linear in time and space such that only a linear algebraic system needs to be solved at each time marching step making it extremely efficient also the existence and uniqueness of numerical solutions are guaranteed for any time step size in addition the scheme is unconditionally energy stable with second order accuracy in time and spectral accuracy in space such that relatively large temporal and spatial mesh sizes can be used to obtain reliable numerical solutions the rigorous proofs for the unconditional energy stable property and solution existence and uniqueness are given furthermore we present several numerical examples to test the proposed numerical algorithm and illustrate its accuracy and efficiency the differences of coarsening dynamics between the cahn hilliard equation and the cahn hilliard navier stokes equations have been investigated as well

preconditioned steepest descent methods for some nonlinear elliptic equations involving p laplacian terms abstract we describe and analyze preconditioned steepest descent psd solvers for fourth and sixth order nonlinear elliptic equations that include p laplacian terms on periodic domains in 2 and 3 dimensions the highest and lowest order terms of the equations are constant coefficient positive linear operators which suggests a natural preconditioning strategy such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena in particular liquid crystals thin film epitaxial growth and phase transformations the analyses of the schemes involve the characterization of the strictly convex energies associated with the equations we first give a general framework for psd in hilbert spaces based on certain reasonable assumptions of the linear pre conditioner a geometric convergence rate is shown for the nonlinear psd iteration we then apply the general theory to the fourth and sixth order problems of interest making use of sobolev embedding and regularity results to confirm the appropriateness of our pre conditioners for the regularized p lapacian problems our results include a sharper theoretical convergence result for p laplacian systems compared to what may be found in existing works we demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods numerical simulations for some important physical application problems including thin film epitaxy with slope selection and the square phase field crystal model are carried out to verify the efficiency of the scheme

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

on ergodic and mixing properties of the triangle map abstract in this paper we study in detail both analytically and numerically the dynamical properties of the triangle map a piecewise parabolic automorphism of the two dimensional torus for different values of the two independent parameters defining the map the dynamics is studied numerically by means of two different symbolic encoding schemes both relying on the fact that it maps polygons to polygons in the first scheme we consider dynamically generated partitions made out of suitable sets of disjoint polygons in the second we consider the standard binary partition of the torus induced by the discontinuity set these encoding schemes are studied in detail and shown to be compatible although not equivalent the ergodic properties of the triangle map are then investigated in terms of the markov transition matrices associated to the above schemes and furthermore compared to the spectral properties of the koopman operator in l 2 t 2 finally a stochastic version of the triangle map is introduced and studied a simple heuristic analysis of the latter yields the correct statistical and scaling behaviours of the correlation functions of the original map

an efficient unconditionally energy stable local discontinuous galerkin scheme for the cahn hilliard brinkman system in this paper we present an efficient and unconditionally energy stable fully discrete local discontinuous galerkin ldg method for approximating the cahn hilliard brinkman chb system which is comprised of a cahn hilliard type equation and a generalized brinkman equation modeling fluid flow the semi discrete energy stability of the ldg method is proved firstly due to the strict time step restriction t o x 4 of explicit time discretization methods for stability we introduce a semi implicit scheme which consists of the implicit euler method combined with a convex splitting of the discrete cahn hilliard energy strategy for the temporal discretization the unconditional energy stability of this fully discrete convex splitting scheme is also proved obviously the fully discrete equations at the implicit time level are nonlinear and to enhance the efficiency of the proposed approach the nonlinear full approximation scheme fas multigrid method has been employed to solve this system of algebraic equations we also show the nearly optimal complexity numerically numerical experiments based on the overall solution method of combining the proposed ldg method convex splitting scheme and the nonlinear multigrid solver are given to validate the theoretical results and to show the effectiveness of the proposed approach for the chb system

numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach summary we present two accurate and efficient numerical schemes for a phase field dendritic crystal growth model which is derived from the variation of a free energy functional consisting of a temperature dependent bulk potential and a conformational entropy with a gradient dependent anisotropic coefficient we introduce a novel invariant energy quadratization approach to transform the free energy functional into a quadratic form by introducing new variables to substitute the nonlinear transformations based on the reformulated equivalent governing system we develop a first and a second order semi discretized scheme in time for the system in which all nonlinear terms are treated semi explicitly the resulting semi discretized equations consist of a linear elliptic equation system at each time step where the coefficient matrix operator is positive definite and thus the semi discrete system can be solved efficiently we further prove that the proposed schemes are unconditionally energy stable convergence test together with 2d and 3d numerical simulations for dendritic crystal growth are presented after the semi discrete schemes are fully discretized in space using the finite difference method to demonstrate the stability and the accuracy of the proposed schemes copyright 2016 john wiley sons ltd

error analysis of a mixed finite element method for a cahn hilliard hele shaw system we present and analyze a mixed finite element numerical scheme for the cahn hilliard hele shaw equation a modified cahn hilliard equation coupled with the darcy flow law this numerical scheme was first reported in feng and wise siam j numer anal 50 1320 1343 2012 with the weak convergence to a weak solution proven in this article we provide an optimal rate error analysis a convex splitting approach is taken in the temporal discretization which in turn leads to the unique solvability and unconditional energy stability instead of the more standard ell infty 0 t l 2 cap ell 2 0 t h 2 l 0 t l2 l2 0 t h2 error estimate we perform a discrete ell infty 0 t h 1 cap ell 2 0 t h 3 l 0 t h1 l2 0 t h3 error estimate for the phase variable through an l 2 l2 inner product with the numerical error function associated with the chemical potential as a result an unconditional convergence for the time step tau in terms of the spatial resolution h is derived the nonlinear analysis is accomplished with the help of a discrete gagliardo nirenberg type inequality in the finite element space gotten by introducing a discrete laplacian delta h h of the numerical solution such that delta h phi in s h h sh for every phi in s h sh where s h sh is the finite element space

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

arithmetical properties of strongly chaotic motions abstract the orbits of the generalized arnold sinai cat maps or hyperbolic automorphisms of the two dimensional torus typify purely chaotic anosov motion we transform the dynamics of the periodic orbits of these maps into modular arithmetic in suitable domains of quadratic integers classify all periodic orbits and show how to determine their periods and initial conditions the methods are based on ideal theory in quadratic fields which is reviewed it is shown that the structure of orbits rests upon some basic arithmetical notions such as unique factorization into prime ideals

an algebraic geometric approach to integrable maps of the plane we show that the dynamics of a birational map on an elliptic curve over a field is typically conjugate to addition by a point under the associated group law when the field is taken to be the function field of rational complex functions of one variable this amounts to an algebraic geometric version of the arnold liouville integrability theorem for planar integrable maps by products of this approach are that birational maps preserving foliations are necessarily the composition of two involutions and that relationships between birational maps preserving the same foliation can be described in terms of the respective points they add on the corresponding weierstrass curves when the result is applied to finite fields it helps explain some universal features of the periodic orbit distribution function for the reductions of integrable maps

a diffuse interface model and semi implicit energy stable finite element method for two phase magnetohydrodynamic flows abstract in this paper we propose a diffuse interface model and finite element approximation for two phase magnetohydrodynamic mhd flows with different viscosities and electric conductivities an energy stable scheme which is based on the finite element method for the spatial discretization and first order semi implicit scheme combined with convex splitting method for the temporal discretization is proposed to solve this new model the numerical scheme is proved to be mass conservative and energy law preserving by leray schauder fixed point theorem the existence of solutions to the numerical scheme is shown the uniqueness of the numerical solutions is obtained utilizing the stability of the numerical scheme and the compactness method the existence of the weak solutions to the two phase mhd model is established as well furthermore given more regularity on the weak solution the convergence of the numerical scheme is derived finally numerical experiments are provided to verify the theoretical results and validate the proposed model

analysis of a novel finite element method for a modified cahn hilliard hele shaw system abstract in this paper a novel finite element method for solving a modified cahn hilliard hele shaw system is proposed the time discretization is based on the convex splitting of the energy functional in the modified cahn hilliard equation i e the high order nonlinear term and the linear term in the chemical potential are treated explicitly and implicitly respectively designing in this way leads to solving a linear system at each time step which is much efficient compared to solving a nonlinear system resulting from most existing schemes the proposed scheme is proved to be unconditionally energy stable and optimally convergent for the phase variable numerical results are presented to support our theoretical analysis

restrictions on microstructure we consider the following question given a set of matrices with no rank one connections does it support a nontrivial young measure limit of gradients our main results are these a a young measure can be supported on four incompatible matrices b in two space dimensions a young measure cannot be supported on finitely many incompatible elastic wells c in three or more space dimensions a young measure can be supported on three incompatible elastic wells and d if supports a nontrivial young measure with mean value 0 then the linear span of must contain a matrix of rank one

the regularity of critical points of polyconvex functionals in this paper we are concerned with the question of regularity of critical points for functionals of the type eq1 we construct a smooth strongly polyconvex eq2 and lipschitzian weak solutions eq3 to the corresponding euler lagrange system which are nowhere c1 moreover we show that f can be chosen in such a way that these irregular weak solutions are weak local minimisers

weak mixing for interval exchange transformations and translation flows we prove that a typical interval exchange transformation is either weakly mixing or it is an irrational rotation we also conclude that a typical translation flow on a typical translation surface of genus g 2 with prescribed singularity types is weakly mixing

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

the failure of rank one connections this article is concerned with interface problems for lipschitz mappings f n n and f n n in the half spaces which agree on the common boundary n 1 n n these naturally occur in mathematical models for material microstructures and crystals the task is to determine the relationship between the sets of values of the differentials df and df for some time it has been thought that the polyconvex hulls df pc and df pc satisfy hadamard s jump condition or are at least rank one connected our examples here refute this idea the estimates of the jacobians we obtain in the course of solving the so called monge ampere inequalities seem also to be of independent interest as an application we construct uniformly elliptic systems of first order partial differential equations in the same homotopy class as the familiar cauchy riemann equations for which the unique continuation property fails

incompatible sets of gradients and metastability we give a mathematical analysis of a concept of metastability induced by incompatibility the physical setting is a single parent phase just about to undergo transformation to a product phase of lower energy density under certain conditions of incompatibility of the energy wells of this energy density we show that the parent phase is metastable in a strong sense namely it is a local minimizer of the free energy in an l1 neighbourhood of its deformation the reason behind this result is that due to the incompatibility of the energy wells a small nucleus of the product phase is necessarily accompanied by a stressed transition layer whose energetic cost exceeds the energy lowering capacity of the nucleus we define and characterize incompatible sets of matrices in terms of which the transition layer estimate at the heart of the proof of metastability is expressed finally we discuss connections with experiments and place this concept of metastability in the wider context of recent theoretical and experimental research on metastability and hysteresis

lipschitz regularity for inner variational equations we obtain lipschitz regularity results for a fairly general class of nonlinear first order pdes these equations arise from the inner variation of certain energy integrals even in the simplest model case of the dirichlet energy the inner stationary solutions need not be differ entiable everywhere the lipschitz continuity is the best possible but the proofs even in the dirichlet case turn out to relay on topological arguments the appeal to the inner stationary solutions in this context is motivated by the classical problems of existence and regularity of the energy minimal deformations in the theory of harmonic mappings and certain mathematical models of nonlinear elasticity specifically neo hookian type problems

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

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

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 connected subsets of m2 2 without rank one connections we prove that connected subsets of m 2 2 without rank one connections are lipschitz graphs of mappings from subsets of a fixed two dimensional subspace to its orthogonal complement under a weaker condition that the set does not have rank one connections locally we are able to establish some global results on the set we also establish some results on lipschitz extensions of the functions thus obtained