the bernstein problem for affine maximal hypersurfaces in this paper we prove the validity of the chern conjecture in affine geometry 18 namely that an affine maximal graph of a smooth locally uniformly convex function on two dimensional euclidean space r 2 must be a paraboloid more generally we shall consider the n dimensional case r n showing that the corresponding result holds in higher dimensions provided that a uniform strict convexity condition holds we also extend the notion of affine maximal to non smooth convex graphs and produce a counterexample showing that the bernstein result does not hold in this generality for dimension n 10

approximation schemes for viscosity solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of applications including the calculus of variations control theory and differential games recently m g crandall and p l lions established the correct notion of generalized solutions for these equations this article discusses the convergence of general approximation schemes to this solution and gives under certain hypotheses explicit error estimates these results are then applied to obtain various representations as limits of solutions of general explicit and implicit finite difference schemes with error estimates

smallest scale estimates for the navier stokes equations for incompressible fluids we consider solutions of the navier stokes equations for incompressible fluids in two and three space dimensions we obtain improved estimates in the limit of vanishing viscosity for the fourier coefficients the coefficients decay exponentially fast for wave numbers larger than the square root of the maximum of the velocity gradients divided by the square root of the viscosity this defines the minimum scale the size of the smallest feature in the flow

affine complete locally convex hypersurfaces an open problem in affine geometry is whether an affine complete locally uniformly convex hypersurface in euclidean n 1 space is euclidean complete for n 2 in this paper we give the affirmative answer as an application it follows that an affine complete affine maximal surface in r 3 must be an elliptic paraboloid

high order essentially nonsocillatory schemes for hamilton jacobi equations hamilton jacobi h j equations are frequently encountered in applications e g in control theory and differential games h j equations are closely related to hyperbolic conservation laws in one space dimension the former is simply the integrated version of the latter similarity also exists for the multidimensional case and this is helpful in the design of difference approximations in this paper high order essentially nonoscillatory eno schemes for h j equations are investigated which yield uniform high order accuracy in smooth regions and sharply resolve discontinuities in the derivatives the eno scheme construction procedure is adapted from that for hyperbolic conservation laws the schemes are numerically tested on a variety of one dimensional and two dimensional problems including a problem related to control optimization and high order accuracy in smooth regions good resolution of discontinuities in the derivatives and convergence to viscosity solutions are observed

convergence of numerical schemes for viscosity solutions to integro differential degenerate parabolic problems arising in financial theory we study the numerical approximation of viscosity solutions for integro differential possibly degenerate parabolic problems similar models arise in option pricing to generalize the celebrated black scholes equation when the processes which generate the underlying stock returns may contain both a continuous part and jumps convergence is proven for monotone schemes and numerical tests are presented and discussed

finite difference schemes for incompressible flows in the velocity impulse density formulation we consider finite difference schemes based on the impulse density variable we show that the original velocity impulse density formulation of oseledets is marginally ill posed for the inviscid flow and this has the consequence that some ordinarily stable numerical methods in other formulations become unstable in the velocity impulse density formulation we present numerical evidence of this instability we then discuss the construction of stable finite difference schemes by requiring that at the numerical level the nonlinear terms be convertible to similar terms in the primitive variable formulation finally we give a simplified velocity impulse density formulation which is free of these complications and yet retains the nice features of the original velocity impulse density formulation with regard to the treatment of boundary we present numerical results on this simplified formulation for the driven cavity flow on both the staggered and non staggered grids

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

on the convergence rate of operator splitting for hamilton jacobi equations with source terms we establish a rate of convergence for a semidiscrete operator splitting method applied to hamilton jacobi equations with source terms the method is based on sequentially solving a hamilton jacobi equation and an ordinary differential equation the hamilton jacobi equation is solved exactly while the ordinary differential equation is solved exactly or by an explicit euler method we prove that the l infty error associated with the operator splitting method is bounded by mathcal o delta t where delta t is the splitting or time step this error bound is an improvement over the existing mathcal o sqrt delta t bound due to souganidis nonlinear anal 9 1985 pp 217 257 in the one dimensional case we present a fully discrete splitting method based on an unconditionally stable front tracking method for homogeneous hamilton jacobi equations it is proved that this fully discrete splitting method possesses a linear convergence rate moreover numerical results are presented to illustrate the theoretical convergence results

approximations for viscosity solutions of hamilton jacobi equations with locally varying time and space grids a new monotone finite difference scheme is introduced that approximates viscosity solutions of first order nonlinear hamilton jacobi equations the main feature of the scheme is that it allows for locally varying time and space grids making it ideal for use with adaptive algorithms explicit a priori error estimates are given to establish convergence numerical examples including adaptive mesh refinement examples demonstrate the effectiveness of the proposed scheme

extensions and imbeddings we establish a connection between the sobolev imbedding theorem and the extendability of sobolev functions as applications we give geometric criteria for extendability and give a result on the dependence of the extension property on the exponentp

numerical discretization of the first order hamilton jacobi equation on triangular meshes we present and analyze several ways of discretizing first order hamilton jacobi equations on unstructured meshes we first discuss two godunov type hamiltonians the first one is an extension of a result by bardi and osher where a particular decomposition of the initial condition is assumed and we point out its practical limits the other one arises from a particular decomposition of the hamiltonian despite its complexity this decomposition enables us to construct a lax friedrichs hamiltonian these schemes all share common properties they are consistent monotonic and independent of the geometrical interpretation of the piecewise linear initial condition under these assumptions and classical ones on the mesh we show these schemes are convergent we describe their high order extensions using the eno technique and provide numerical illustrations 1996 john wiley sons inc

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

extremal metrics on toric surfaces a continuity method the paper develops an existence theory for solutions of the abreu equation which include extremal metrics on toric surfaces the technique employed is a continuity method combined with blow up arguments general existence results are obtained assuming a hypothesis the m condition on the solutions which is shown to be related to the injectivity radius

a bernstein theorem for affine maximal type hypersurfaces under decaying convexity abstract in this paper we will study a fourth order fully nonlinear pde u i j d i j w 0 w det d 2 u 0 of affine maximal type by a formula of quasi norm of third derivatives derived from pure pde tricks we will prove that any entire convex solution must be a quadratic function provided 0 1 2 n 1 2 n 8 1 2 n 1 2 n 8 and the minimal convex constant r decays slower than r n for some positive constant n given explicitly below

high resolution nonoscillatory central schemes for hamilton jacobi equations in this paper we construct second order central schemes for multidimensional hamilton jacobi equations and we show that they are nonoscillatory in the sense of satisfying the maximum principle thus these schemes provide the first examples of nonoscillatory second order godunov type schemes based on global projection operators numerical experiments are performed l 1 l infty errors and convergence rates are calculated for convex hamiltonians numerical evidence confirms that our central schemes converge with second order rates when measured in the l1 norm advocated in our recent paper numer math to appear the standard l infty norm however fails to detect this second order rate

sobolev embeddings extensions and measure density condition there are two main results in the paper in the first one theorem 1 we prove that if the sobolev embedding theorem holds in in any of all the possible cases then satisfies the measure density condition the second main result theorem 5 provides several characterizations of the wm p extension domains for 1 p as a corollary we prove that the property of being a w1 p extension domain 1 p is invariant under bi lipschitz mappings theorem 8

a class of high resolution difference schemes for nonlinear hamilton jacobi equations with varying time and space grids based on a simple projection of the solution increments of the underlying partial differential equations pdes at each local time level this paper presents a difference scheme for nonlinear hamilton jacobi h j equations with varying time and space grids the scheme is of good consistency and monotone under a local cfl type condition moreover one may deduce a conservative local time step scheme similar to osher and sanders scheme approximating hyperbolic conservation law cl from our scheme according to the close relation between cls and h j equations second order accurate schemes are constructed by combining the reconstruction technique with a second order accurate runge kutta time discretization scheme or a lax wendroff type method they keep some good properties of the global time step schemes including stability and convergence and can be applied to solve numerically the initial boundary value problems of viscous h j equations they are also suitable to parallel computing numerical errors and the experimental rate of convergence in the lp norm p 1 2 and infty are obtained for several one and two dimensional problems the results show that the present schemes are of higher order accuracy

bernstein problem of affine maximal type hypersurfaces on dimension n 3 abstract bernstein problem for affine maximal type equation 0 1 u i j d i j w 0 w det d 2 u x r n has been a core problem in affine geometry a conjecture proposed firstly by chern 1977 6 for entire graph and then extended by trudinger wang 2000 14 to its fully generality asserts that any euclidean complete affine maximal type locally uniformly convex c 4 hypersurface in r n 1 must be an elliptic paraboloid at the same time this conjecture was solved completely by trudinger wang for dimension n 2 and 3 4 and later extended by jia li 2009 12 to n 2 3 4 1 see also zhou 2012 16 for a different proof on the past twenty years many efforts were done toward higher dimensional issues but not really successful yet even for the case of dimension n 3 in this paper we will construct non quadratic affine maximal type hypersurfaces which are euclidean compete for n 3 1 2 n 1 n

l 1 stability and error estimates for approximate hamilton jacobi solutions we study the l 1 stability and error estimates of general approximate solutions for the cauchy problem associated with multidimensional hamilton jacobi h j equations for strictly convex hamiltonians we obtain a priori error estimates in terms of the truncation errors and the initial perturbation errors we then demonstrate this general theory for two types of approximations approximate solutions constructed by the vanishing viscosity method and by godunov type finite difference methods if we let epsilon denote the small scale of such approximations the viscosity amplitude epsilon the spatial grad size delta x etc then our l 1 error estimates are of cal o epsilon and are sharper than the classical l infty results of order one half cal o sqrt epsilon the main building blocks of our theory are the notions of the semi concave stability condition and l 1 measure of the truncation error the whole theory could be viewed as a multidimensional extension of the lip prime stability theory for one dimensional nonlinear conservation laws developed by tadmor et al 34 24 25 in addition we construct new godunov type schemes for h j equations which consist of an exact evolution operator and a global projection operator here we restrict our attention to linear projection operators first order schemes we note however that our convergence theory applies equally well to nonlinear projections used in the context of modern high resolution conservation laws we prove semi concave stability and obtain l 1 bounds on their associated truncation errors l 1 convergence of order one then follows second order central godunov type schemes are also constructed numerical experiments are performed errors and orders are calculated to confirm our l 1 theory

finite volume schemes for hamilton jacobi equations we introduce two classes of monotone finite volume schemes for hamilton jacobi equations the corresponding approximating functions are piecewise linear defined on a mesh consisting of triangles the schemes are shown to converge to the viscosity solution of the hamilton jacobi equation

besov spaces on domains in rd we study besov spaces 5 lp i2 0 p q a oo on do mains i2 in rd we show that there is an extension operator w which is a bounded mapping from b lp u onto b lp rd this is then used to derive various properties of the besov spaces such as interpolation theorems for a pair of b lp il atomic decompositions for the elements of 5 lp f2 and a description of the besov spaces by means of spline approximation

unconditionally stable methods for hamilton jacobi equations we present new numerical methods for constructing approximate solutions to the cauchy problem for hamilton jacobi equations of the form ut h dxu 0 the methods are based on dimensional splitting and front tracking for solving the associated non strictly hyperbolic system of conservation laws pt dx h p 0 where p dx u in particular our methods depend heavily on a front tracking method for one dimensional scalar conservation laws with discontinuous coefficients the proposed methods are unconditionally stable in the sense that the time step is not limited by the space discretization and they can be viewed as large time step godunov type or front tracking methods we present several numerical examples illustrating the main features of the proposed methods we also compare our methods with several methods from the literature

alternating evolution discontinuous galerkin methods for hamilton jacobi equations in this work we propose a high resolution alternating evolution discontinuous galerkin aedg method to solve hamilton jacobi equations the construction of the aedg method is based on an alternating evolution system of the hamilton jacobi equation following the previous work liu et al 2013 31 on ae schemes for hamilton jacobi equations a semi discrete aedg scheme derives directly from a sampling of this system on alternating grids higher order accuracy is achieved by a combination of high order polynomial approximation near each grid and a time discretization with matching accuracy the aedg methods have the advantage of easy formulation and implementation and efficient computation of the solution for the linear equation we prove the l 2 stability of the method numerical experiments for a set of hamilton jacobi equations are presented to demonstrate both accuracy and capacity of these aedg schemes

new high resolution semi discrete central schemes for hamilton jacobi equations we introduce a new high resolution central scheme for multidimensional hamilton jacobi equations the scheme retains the simplicity of the non oscillatory central schemes developed by c t lin and e tadmor in press siam j sci comput yet it enjoys a smaller amount of numerical viscosity independent of 1 t by letting t 0 we obtain a new second order central scheme in the particularly simple semi discrete form along the lines of the new semi discrete central schemes recently introduced by the authors in the context of hyperbolic conservation laws fully discrete versions are obtained with appropriate runge kutta solvers the smaller amount of dissipation enables efficient integration of convection diffusion equations where the accumulated error is independent of a small time step dictated by the cfl limitation the scheme is non oscillatory thanks to the use of nonlinear limiters here we advocate the use of such limiters on second discrete derivatives which is shown to yield an improved high resolution when compared to the usual limitation of first derivatives numerical experiments demonstrate the remarkable resolution obtained by the proposed new central scheme

numerical schemes for conservation laws via hamilton jacobi equations we present some difference approximation schemes which converge to the entropy solution of a scalar conservation law having a convex flux the numerical methods described here take their origin from approximation schemes for hamilton jacobi bellman equations related to optimal control problems and exhibit several interesting features the convergence result still holds for quite arbitrary time steps the main assumption for convergence can be interpreted as a discrete analogue of oleinik s entropy condition numerical diffusion around the shocks is very limited some tests are included in order to compare the performances of these methods with other classical methods godunov tvd