a legendre galerkin spectral method for optimal control problems governed by elliptic equations in this paper we study the legendre galerkin spectral approximation for a constrained optimal control problem we first derive a priori error estimates for the spectral approximation of optimal co

on the master symmetries and bi hamiltonian structure of the toda lattice starting from a conformal symmetry higher order poisson tensors deformation relations and master symmetries for the toda lattice are obtained a hierarchy of time dependent symmetries is also constructed using reduction deformation relations previously known to hold up to a certain equivalence relation are shown to be exact

almost optimal local well posedness for the 3 1 dimensional maxwell klein gordon equations this paper contains a detailed study of the local in time regularity properties of the maxwell klein gordon mkg equations the mkg equations represent a physical model for the interaction of a spin 0 particle sometimes referred to as a higgs or scalar field with the classical equations of electrodynamics i e maxwell s equations in terms of the calculus of variations the mkg system consists of the euler lagrange equations associated to the density l 4 f f 1 2 d d where f a a is the electro magnetic field associated to the set a of real valued potentials on the minkowski space d 1a is the corresponding covariant derivative with respect to those potentials and r c is a complex scalar field in terms of the field f the euler lagrange equations read f d 1a d d 0 1b

a legendre galerkin spectral method for optimal control problems governed by stokes equations in this paper we study the legendre galerkin spectral approximation of distributed optimal control problems governed by stokes equations we show that the discretized control problems satisfy the well known babuska brezzi conditions by choosing an appropriate pair of discretization spaces for the velocity and the pressure constructing suitable base functions of the discretization spaces leads to sparse coefficient matrices we first derive a priori error estimates in both h 1 and l 2 norms for the legendre galerkin approximation of the unconstrained control problems then both a priori and a posteriori error estimates are obtained for control problems with the constraints of an integral type thanks to the higher regularity of the optimal control finally some illustrative numerical examples are presented to demonstrate the error estimates

convergence of a finite element approximation to a state constrained elliptic control problem we consider an elliptic optimal control problem with pointwise state constraints the cost functional is approximated by a sequence of functionals which are obtained by discretizing the state equation with the help of linear finite elements and enforcing the state constraints in the nodes of the triangulation the corresponding minima are shown to converge in l 2 to the exact control as the discretization parameter tends to zero furthermore error bounds for the control and the state are obtained in both two and three space dimensions finally we present numerical examples which confirm our analytical findings

new cartesian grid methods for interface problems using the finite element formulation new finite element methods based on cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients the triangulations in these methods do not need to fit the interfaces the basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximately both non conforming and conforming finite element spaces are considered corresponding interpolation functions are proved to be second order accurate in the maximum norm the conforming finite element method has been shown to be convergent with cartesian triangulations these new methods can be used as finite difference methods numerical examples are provided to support the methods and the theoretical analysis

optimal control of semilinear elliptic equations in measure spaces optimal control problems in measure spaces governed by semilinear elliptic equations are considered first order optimality conditions are derived and structural properties of their solutions in particular sparsity are discussed necessary and sufficient second order optimality conditions are obtained as well on the basis of the sufficient conditions stability of the solutions is analyzed highly nonlinear terms can be incorporated by utilizing an l infty omega regularity result for solutions of the first order necessary optimality conditions

adaptive finite element approximation for distributed elliptic optimal control problems in this paper sharp a posteriori error estimators are derived for a class of distributed elliptic optimal control problems these error estimators are shown to be useful in adaptive finite element approximation for the optimal control problems and are implemented in the adaptive approach our numerical results indicate that the sharp error estimators work satisfactorily in guiding the mesh adjustments and can save substantial computational work

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

long time existence for small amplitude semilinear wave equations we prove that a certain class of semilinear wave equations has global solutions if the initial data is small these existence results rely on mixed norm angular radial space time estimates more specifically we consider power nonlinearities u f p u where f p u u p and 2 t denotes the d alembertian on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i in 1979 john investigated this equation for small initial data in dimension n 3 he proved that if p 1 2 then there is a global solution whereas for p n dimensions and the critical power p c above which global solutions exist should be the positive root of n 1 p 2 c n 1 p c 2 0 sideris proved that for all n the solutions can blow up if p p c recently zhou proved that for n 4 one has global solutions if p p c 2 here we prove that for n 8 one has global solutions if p p c furthermore we proved that for all n one has global solutions if p p c and the initial data are spherically symmetric in the radial case we use l p estimates derived from the explicit form of the fundamental solution using inequalities from classical analysis in the nonradial case we write the fundamental solution in polar coordinates and use fourier integral and special function estimates to handle the angular part

a high accuracy nfem for constrained optimal control problems governed by elliptic equations in this paper a new nonconforming finite element method nfem is proposed for the constrained optimal control problems ocps governed by elliptic equations the state and co state are approximated by the nonconforming eq 1 rot element and the control is approximated by the piecewise constant element respectively the superclose properties for the above three variables are obtained moreover for the state and co state the superconvergence results are achieved in a broken energy norm by using the post processing technique lastly some numerical examples are provided to verify the theoretical analysis

control of an elliptic problem with pointwise state constraints this paper deals with a quadratic control problem for elliptic equations with pointwise state constraints existence and uniqueness of the solution is proved optimality conditions are given and regularity of the optimal solution is investigated

reduced basis method for parametrized elliptic optimal control problems we propose a suitable model reduction paradigm the certied reduced basis method rb for the rapid and reliable solution of parametrized optimal control problems governed by partial dierential equations pdes in particular we develop the methodology for parametrized quadratic optimization problems with elliptic equations as constraint and infinite dimensional control variable firstly we recast the optimal control problem in the framework of saddle point problems in order to take advantage of the already developed rb theory for stokes type problems then the usual ingredients of the rb methodology are called into play a galerkin projection onto a low dimensional space of basis functions properly selected by an adaptive procedure an affine parametric dependence enabling to perform competitive offline online splitting in the computational procedure an efficient and rigorous a posteriori error estimate on the state control and adjoint variables as well as on the cost functional finally we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique

a bilinear estimate with applications to the kdv equation u x 0 u0 x where u0 h r our principal aim here is to lower the best index s for which one has local well posedness in h r i e existence uniqueness persistence and continuous dependence on the data for a finite time interval whose size depends on u0 hs equation in 1 1 was derived by korteweg and de vries 21 as a model for long wave propagating in a channel a large amount of work has been devoted to the existence problem for the ivp 1 1 for instance see 9 10 the inverse scattering method applies to this problem and under appropriate decay assumptions on the data several existence results have been established see 5 6 14 28 33 another approach inherited from hyperbolic problems relies on the energy estimates and in particular shows that 1 1 is locally well posed in h r for s 3 2 see 2 3 12 29 30 31 using these results and conservation laws global in time well posedness in h r s 2 was established see 3 12 30 also global in time weak solutions in the energy space h r were constructed in 34 in 13 and 22 a local smoothing effect for solutions of 1 1 was discovered this combined with the conservation laws was used in 13 and 22 to construct global in time weak solutions with data in h r and even in l r in 16 we introduced oscillatory integral techniques to establish local well posedness of 1 1 in h r s 3 4 and hence global in time well posedness in h r s 1 in 16 we showed how to obtain the above mentioned result by picard iteration in an appropriate function space in 4 j bourgain introduced new function spaces adapted to the linear operator t 3 x for which there are good bilinear estimates for the nonlinear term x u 2 using these spaces bourgain was able to establish local well posedness of 1 1 in h r l r and hence by a conservation

finite element methods and their convergence for elliptic and parabolic interface problems in this paper we consider the finite element methods for solving second order elliptic and parabolic interface problems in two dimensional convex polygonal domains nearly the same optimal l 2 norm and energy norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth though the regularities of the solutions are low on the whole domain the assumptions on the finite element triangulation are reasonable and practical

immersed finite elements for optimal control problems of elliptic pdes with interfaces this paper presents a numerical method and analysis based on the variational discretization concept for optimal control problems governed by elliptic pdes with interfaces the method uses a simple uniform mesh which is independent of the interface due to the jump of the coefficient across the interface the standard linear finite element method cannot achieve optimal convergence when the uniform mesh is used therefore the immersed finite element method ifem developed in li et al 20 is used to discretize the state equation required in the variational discretization approach optimal error estimates for the control state and adjoint state are derived numerical examples are provided to confirm the theoretical results

explosion for some semilinear wave equations in this note we show that known results on the propagation of singularities for semilinear wave equations suffice for the construction of explosive solutions with arbitrarily small data in particular for the quadratically nonlinear wave equation in one space dimension utt uxx au bup cu 3 q u u 1 we show that the only nonlinearity without such explosions is nirenberg s equation b 0 c a which is transformed to the linear wave equation by the substitution u ea this nonlinearity is also singled out by special compactness properties 3 and its role in the equipartition of energy cl another much studied equation for which we supply explosive solutions is the carleman model of transport theory 8 s 8 f 8 a u r u 242

the wave maps equation the wave maps equation has become a very popular topic in recent years the aim of these expository notes is to present a non technical survey of the ideas and methods which have proved useful in the study of wave maps leading up to the latest results the remaining open problems are also stated and explained

optimal solvers for pde constrained optimization optimization problems with constraints which require the solution of a partial differential equation arise widely in many areas of the sciences and engineering particularly in problems of design the solution of such pde constrained optimization problems is usually a major computational task here we consider simple problems of this type distributed control problems in which the 2 and 3 dimensional poisson problem is the pde the large dimensional linear systems which result from discretization and which need to be solved are of saddle point type we introduce two optimal preconditioners for these systems which lead to convergence of symmetric krylov subspace iterative methods in a number of iterations which does not increase with the dimension of the discrete problem these preconditioners are block structured and involve standard multigrid cycles the optimality of the preconditioned iterative solver is proved theoretically and verified computationally in several test cases the theoretical proof indicates that these approaches may have much broader applicability for other pdes

quadratic forms for the 1 d semilinear schrodinger equation this paper is concerned with 1 d quadratic semilinear schr6dinger equations we study local well posedness in classical sobolev space hs of the associated initial value problem and periodic boundary value problem our main interest is to obtain the lowest value of s which guarantees the desired local well posedness result we prove that at least for the quadratic cases these values are negative and depend on the structure of the nonlinearity considered

superconvergence properties of optimal control problems an optimal control problem for a two dimensional 2 d elliptic equation is investigated with pointwise control constraints this paper is concerned with discretization of the control by piecewise constant functions the state and the adjoint state are discretized by linear finite elements approximations of the optimal solution of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state it is proved that these approximations have convergence order h2

hs global well posedness for semilinear wave equations abstract we consider the cauchy problem for semilinear wave equations in r n with n 3 making use of bourgain s method in conjunction with the endpoint strichartz estimates of keel and tao we establish the hs global well posedness with s