slow fast n dimensional piecewise linear differential systems abstract in this article we analyze n dimensional slow fast systems in a piecewise linear framework in particular we prove a fenichel s like theorem where we give an explicit expression for the invariant slow manifold that leads to the proof of the existence and location of maximal canards orbits we show that these orbits perturb from singular orbits through contact points of order greater than or equal to two between the reduced flow and the fold manifold in the particular case n 3 we show that the unique contact point is a visible two fold singularity

feasible and noninterior path following in constrained minimization with low multiplier regularity primal dual path following methods for constrained minimization problems in function space with low multiplier regularity are introduced and analyzed regularity properties of the path are proved the path structure allows us to define approximating models which are used for controlling the path parameter in an iterative process for computing a solution of the original problem the moreau yosida regularized subproblems of the new path following technique are solved efficiently by semismooth newton methods the overall algorithmic concept is provided and numerical tests including a comparison with primal dual path following interior point methods for state constrained optimal control problems show the efficiency of the new concept

sufficient second order optimality conditions for semilinear control problems with pointwise state constraints second order sufficient optimality conditions are established for the optimal control of semilinear elliptic and parabolic equations with pointwise constraints on the control and the state in contrast to former publications on this subject the cone of critical directions is the smallest possible in the sense that the second order sufficient conditions are the closest to the associated necessary ones the theory is developed for elliptic distributed controls in domains up to dimension three moreover problems of elliptic boundary control and parabolic distributed control are discussed in spatial domains of dimension two and one respectively

an analytic expression for the electronic correlation term of the kinetic functional we propose an analytic formula for the non local fisher information functional or electronic kinetic correlation term appearing in the expression of the kinetic density functional such an explicit formula is constructed on the basis of well founded physical arguments and a rigorous mathematical prescription

maximal parabolic regularity for divergence operators including mixed boundary conditions abstract we show that elliptic second order operators a of divergence type fulfill maximal parabolic regularity on distribution spaces even if the underlying domain is highly non smooth the coefficients of a are discontinuous and a is complemented with mixed boundary conditions applications to quasilinear parabolic equations with non smooth data are presented

canards in piecewise linear systems explosions and super explosions we show that a planar slow fast piecewise linear pwl system with three zones admits limit cycles that share a lot of similarity with van der pol canards in particular an explosive growth using phase space compactification we show that these quasi canard cycles are strongly related to a bifurcation at infinity furthermore we investigate a limiting case in which we show the existence of a continuum of canard homoclinic connections that coexist for a single parameter value and with amplitude ranging from an order of e to an order of 1 a phenomenon truly associated with the non smooth character of this system and which we call super explosion

optimal control of the thermistor problem in three spatial dimensions part 2 optimality conditions this paper is concerned with the state constrained optimal control of the three dimensional thermistor problem a fully quasilinear coupled system of a parabolic and elliptic pde with mixed boundary conditions this system models the heating of a conducting material by means of direct current local existence uniqueness and continuity for the state system as well as existence of optimal solutions admitting global in time solutions to the optimization problem were shown in the the companion paper of this work in this part we address further properties of the set of controls whose associated solutions exist globally such as openness which includes analysis of the linearized state system via maximal parabolic regularity the adjoint system involving measures is investigated using a duality argument these results allow us to derive first order necessary conditions for the optimal control problem in the form of a qualified optimality system in which we do not need to refer to the set of controls admit

optimal control of a convective boundary condition in a thermistor problem we consider the optimal control of a two dimensional steady state thermistor the problem is described by a system of two nonlinear elliptic partial differential equations with appropriate boundary conditions which model the coupling of the thermistor to its surroundings based on physical considerations an objective functional to be minimized is introduced and the convective boundary coefficient is taken as the control existence and uniqueness of the optimal control are proven to characterize this optimal control the optimality system consisting of the state and adjoint equations is derived

second order sufficient optimality conditions for parabolic optimal control problems with pointwise state constraints in this paper we study optimal control problems governed by semilinear parabolic equations where the spatial dimension is two or three moreover we consider pointwise constraints on the control and on the state we formulate first order necessary and second order sufficient optimality conditions we make use of recent results regarding elliptic regularity and apply the concept of maximal parabolic regularity to the occurring partial differential equations

a nonlinear conjugate gradient method with a strong global convergence property conjugate gradient methods are widely used for unconstrained optimization especially large scale problems the strong wolfe conditions are usually used in the analyses and implementations of conjugate gradient methods this paper presents a new version of the conjugate gradient method which converges globally provided the line search satisfies the standard wolfe conditions the conditions on the objective function are also weak being similar to those required by the zoutendijk condition

optimal control of the thermistor problem in three spatial dimensions part 1 existence of optimal solutions this paper is concerned with the state constrained optimal control of the three dimensional thermistor problem a fully quasi linear coupled system of a parabolic and elliptic pdes with mixed boundary conditions this system models the heating of a conducting material by means of direct current local existence uniqueness and continuity for the state system are derived by employing maximal parabolic regularity in the fundamental theorem of pruss global solutions and controls admitting such are addressed and existence of optimal controls is shown if the temperature gradient is under control this work is the first of two papers on the three dimensional thermistor problem

relativistic schrodinger operators asymptotic behavior of the eigenfunctions nonrelativistic schrodinger operators are perturbations of the negative laplacian and the connection with stochastic processes and brownian motion in particular is well known and usually goes under the name of feynman and kac we present a similar connection between a class of relativistic schrodinger operators and a class of processes with stationary independent increments in particular we investigate the decay of the eigenfunctions of these operators and we show that not only exponential decay but also polynomial decay can occur

universal constraints on the location of extrema of eigenfunctions of non local schrodinger operators abstract we derive a lower bound on the location of global extrema of eigenfunctions for a large class of non local schrodinger operators in convex domains under dirichlet exterior conditions featuring the symbol of the kinetic term the strength of the potential and the corresponding eigenvalue and involving a new universal constant we show a number of probabilistic and spectral geometric implications and derive a faber krahn type inequality for non local operators our study also extends to potentials with compact support and we establish bounds on the location of extrema relative to the boundary edge of the support or level sets around minima of the potential

analysis and finite element approximation of an optimal control problem for the oseen viscoelastic fluid flow in this article we study a boundary control problem for an oseen type model of viscoelastic fluid flow the existence of a unique optimal solution is proved and an optimality system is derived by the first order necessary condition we investigate finite element approximations to a solution of the optimality system and a solution algorithm for the system based on the gradient method

canards in r3 abstract we give a geometric analysis of canard solutions in three dimensional singularly perturbed systems with a folded two dimensional critical manifold by analysing the reduced flow we obtain singular canard solutions passing through a singularity on the fold curve we classify these singularities called canard points as folded saddles folded nodes and folded saddle nodes we prove the existence of canard solutions in the case of the folded saddle we show the existence of canards in the folded node case provided a generic non resonance condition is satisfied and in a subcase of the folded saddle node the proof is based on the blow up method

optimal control of a nonlocal thermistor problem with abc fractional time derivatives abstract we study an optimal control problem associated to a fractional nonlocal thermistor problem involving the abc atangana baleanu caputo fractional time derivative we first prove the existence and uniqueness of solution then we show that an optimal control exists moreover we obtain the optimality system that characterizes the control

a simple proof of hardy lieb thirring inequalities we give a short and unified proof of hardy lieb thirring inequalities for moments of eigenvalues of fractional schrodinger operators the proof covers the optimal parameter range it is based on a recent inequality by solovej sorensen and spitzer moreover we prove that any non magnetic lieb thirring inequality implies a magnetic lieb thirring inequality with possibly a larger constant

numerical identification of a robin coefficient in parabolic problems this paper studies a regularization approach for an inverse problem of estimating a spatially and temporally dependent robin coefficient arising in the analysis of convective heat transfer the parameter to state map is analyzed especially a differentiability result is established a regularization approach is proposed and the properties e g existence and optimality system of the functional are investigated a finite element method is adopted for discretizing the continuous optimization problem and the convergence of the finite element approximations as the mesh size and temporal step size tend to zero is established numerical results by the conjugate gradient method for oneand two dimensional problems are presented

numerical approximation of a quasi newtonian stokes flow problem with defective boundary conditions in this article we study the numerical approximation of a quasi newtonian stokes flow problem where only the flow rates are specified at the inflow and outflow boundaries a variational formulation of the problem using lagrange multipliers to enforce the stated flow rates is given the existence and the uniqueness to the continuous and discrete variational formulations of the solution are shown an error analysis for the numerical approximation is also given numerical computations are included which demonstrate the approximation scheme studied

remarks on the existence uniqueness and semi explicit solvability of systems of autonomous partial differential equations in divergence form with constant boundary conditions a method is proposed for reducing autonomous nonlinear boundary value problems of partial differential equations with a particular structure and constant boundary conditions to the simple problem this requires the solution of a non standard two point system of ordinary differential equations some applications are also presented for classical problems of mathematical physics

finite element approximation of viscoelastic fluid flow existence of approximate solutions and error bounds we study a finite element approximation of viscoelastic fluid flow obeying an oldroyd b type constitutive law the approximate stress velocity and pressure are respectivelyp 1 discontinuous p 2 continuous p 1 continuous we use the method of lesaint raviart for the convection of the extra stress tensor we suppose that the continuous problem admits a sufficiently smooth and sufficiently small solution we show by a fixed point method that the approximate problem has a solution and we give an error bound

fractional hardy lieb thirring and related inequalities for interacting systems we prove analogues of the lieb thirring and hardy lieb thirring inequalities for many body quantum systems with fractional kinetic operators and homogeneous interaction potentials where no anti symmetry on the wave functions is assumed these many body inequalities imply interesting one body interpolation inequalities and we show that the corresponding one and many body inequalities are actually equivalent in certain cases

on maximal parabolic regularity for non autonomous parabolic operators abstract we consider linear inhomogeneous non autonomous parabolic problems associated to sesquilinear forms with discontinuous dependence of time we show that for these problems the property of maximal parabolic regularity can be extrapolated to time integrability exponents r 2 this allows us to prove maximal parabolic l r regularity for discontinuous non autonomous second order divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations

discrete maximal parabolic regularity for galerkin finite element methods the main goal of the paper is to establish time semidiscrete and space time fully discrete maximal parabolic regularity for the time discontinuous galerkin solution of linear parabolic equations such estimates have many applications they are essential for example in establishing optimal a priori error estimates in non hilbertian norms without unnatural coupling of spatial mesh sizes with time steps

second order sufficient optimality conditions for some state constrained control problems of semilinear elliptic equations this paper deals with a class of optimal control problems governed by elliptic equations with nonlinear boundary condition the case of boundary control is studied pointwise constraints on the control and certain equality and set constraints on the state are considered second order sufficient conditions for local optimality of controls are established

fisher information kinetic energy and uncertainty relation inequalities by interpolating between fisher information and mechanical kinetic energy we introduce a general notion of kinetic energy with respect to a parameter of schrodinger wavefunctions from a statistical inference perspective kinetic energy is the sum of fisher information and an integral of a parametrized analogue of quantum mechanical current density related to phase a family of integral inequalities concerning kinetic energy and moments are established among which the cramer rao inequality and the weyl heisenberg inequality are special cases in particular the integral inequalities involving the negative order moments are relevant to the study of electron systems moreover by specifying the parameter to a scale we obtain a family of inequalities of uncertainty relation type which incorporate the position and momentum observables symmetrically in a single quantity

minimizers for the hartree fock bogoliubov theory of neutron stars and white dwarfs we prove the existence of minimizers for hartree fock bogoliubov hfb energy functionals with attractive two body interactions given by newtonian gravity this class of hfb functionals serves as a model problem for self gravitating relativistic fermi systems which are found in neutron stars and white dwarfs furthermore we derive some fundamental properties of hfb minimizers such as a decay estimate for the minimizing density a decisive feature of the hfb model in gravitational physics is its failure of weak lower semicontinuity this fact essentially complicates the analysis compared to the well studied hartree fock theories in atomic physics

analysis and approximation of the cross model for quasi newtonian flows with defective boundary conditions a numerical method for a generalized newtonian fluid with flow rate boundary conditions is considered the defect boundary condition problem is formulated as a constrained optimal control problem where a flow balance is forced on the inflow and outflow boundaries using a neumann control the control problem is analyzed for an existence result and the lagrange multiplier rule a decoupling solution algorithm is presented and numerical experiments are provided to validate robustness of the algorithm

energy corrected fem for optimal dirichlet boundary control problems in the presence of re entrant corners the solution of pde constrained optimization problems in general has singular components even when the given data are smooth consequently the use of standard finite element discretization gives rise to the so called pollution effect this means that only a reduced convergence rate compared to the best approximation error is obtained we discuss how optimal convergence rates in weighted norms can be achieved using the idea of energy corrected finite element methods applied to optimal dirichlet boundary control problem in the energy space we present optimal error estimates in weighted norms for the state variable and for the control several numerical examples illustrate the obtained theoretical results

finite element approximation of viscoelastic fluid flow existence of approximate solutions and error bounds continuous approximation of the stress the author studies a finite element approximation of viscoelastic fluid flow obeying an oldroyd b type constitutive law the approximate stress velocity and pressure are respectively p 1 continuous p 2 continuous and p 1 continuous the streamline upwinding petrov galerkin method for the convection of the extra stress tensor is used suppose that the continuous problem admits a sufficiently smooth and sufficiently small solution it is shown by a fixed point method that the approximate problem has a solution and an error bound is given

first and second order optimality conditions for a class of optimal control problems with quasilinear elliptic equations a class of optimal control problems for quasilinear elliptic equations is considered where the coefficients of the elliptic differential operator depend on the state function first and second order optimality conditions are discussed for an associated control constrained optimal control problem main emphasis is laid on second order sufficient optimality conditions to this aim the regularity of the solutions to the state equation and its linearization is studied in detail and the pontryagin maximum principle is derived one of the main difficulties is the nonmonotone character of the state equation

mixed mode oscillations in a piecewise linear system with multiple time scale coupling abstract in this work we analyze a four dimensional slow fast piecewise linear system with three time scales presenting mixed mode oscillations the system possesses an attractive limit cycle along which oscillations of three different amplitudes and frequencies can appear namely small oscillations pulses medium amplitude and one surge largest amplitude in addition to proving the existence and attractiveness of the limit cycle we focus our attention on the canard phenomena underlying the changes in the number of small oscillations and pulses we analyze locally the existence of secondary canards leading to the addition or subtraction of one small oscillation and describe how this change is globally compensated for or not with the addition or subtraction of one pulse

discrete maximal parabolic regularity for galerkin finite element methods for nonautonomous parabolic problems the main goal of the paper is to establish time semidiscrete and space time fully discrete maximal parabolic regularity for the lowest order time discontinuous galerkin solution of linear parabolic

canards folded nodes and mixed mode oscillations in piecewise linear slow fast systems canard induced phenomena have been extensively studied in the last three decades from both the mathematical and the application viewpoints canards in slow fast systems with at least two slow variables especially near folded node singularities give an essential generating mechanism for mixed mode oscillations mmos in the framework of smooth multiple timescale systems there is a wealth of literature on such slow fast dynamical systems and many models displaying canard induced mmos particularly in neuroscience in parallel since the late 1990s several papers have shown that the canard phenomenon can be faithfully reproduced with piecewise linear pwl systems in two dimensions although very few results are available in the three dimensional case the present paper aims to bridge this gap by analyzing canonical pwl systems that display folded singularities primary and secondary canards with a similar control of the maximal winding number as in the smooth case we also show that the singular phas

on enhanced binding and related effectsin the non and semi relativistic pauli fierz models we prove enhanced binding and increase of binding energies in the non and semi relativistic pauli fierz models for arbitrary values of the fine structure constant and the ultra violet cut off and discuss the resulting improvement of exponential localization of ground state eigenvectors for the semi relativistic model we also discuss the increase of the renormalized electron mass and determine the linear leading order term in the asymptotics of the self energy as the ultra violet cut off goes to infinity

analysis of optimal control problems for the two dimensional thermistor system an optimal control problem for the thermistor system is considered first the precise mathematical problem is established and the proof of existence of the optimal solution is given with appropriate function spaces then g teaux differentiability is shown for the thermistor system with respect to control and the optimality system is obtained

on the upper bound of the electronic kinetic energy in terms of density functionals we propose a simple density functional expression for the upper bound of the kinetic energy for electronic systems such a functional is valid in the limit of slowly varying density its validity outside this regime is discussed by making a comparison with upper bounds obtained in the previous work the advantages of the functional proposed for applications to realistic systems are briefly discussed

optimal control of nonlocal thermistor equations we are concerned with the optimal control problem of the well known nonlocal thermistor problem i e in studying the heat transfer in the resistor device whose electrical conductivity is strongly dependent on the temperature existence of an optimal control is proved the optimality system consisting of the state system coupled with adjoint equations is derived together with a characterisation of the optimal control uniqueness of solution to the optimality system and therefore the uniqueness of the optimal control is established the last part is devoted to numerical simulations

residual a posteriori error estimator for a three field model of a non linear generalized stokes problem in this article we propose and analyze an a posteriori error estimator for a three field model of a generalized stokes problem the components of the a posteriori error estimator are defined via a non linear projection of the residues of the variational equations both upper and lower bounds for the approximation error are derived in terms of the components of the a posteriori error estimator the non linear projections do not need to be explicitly computed to construct the a posteriori error estimates

an uncertainty principle for fermions with generalized kinetic energy we derive semiclassical upper bounds for the number of bound states and the sum of negative eigenvalues of the one particle hamiltoniansh f i v x acting onl2 n these bounds are then used to derive a lower bound on the kinetic energy sum limits j 1 n left langle psi f i nabla j psi right rangle for ann fermion wavefunction we discuss two examples in more detail f p p andf p p2 m2 1 2 m both in three dimensions

relaxation oscillation and canard explosion abstract we give a geometric analysis of relaxation oscillations and canard cycles in singularly perturbed planar vector fields the transition from small hopf type cycles to large relaxation cycles which occurs in an exponentially thin parameter interval is described as a perturbation of a family of singular cycles the results are obtained by means of two blow up transformations combined with standard tools of dynamical systems theory the efficient use of various charts is emphasized the results are applied to the van der pol equation

an extension of pontryagin s principle for state constrained optimal control of semilinear elliptic equations and variational inequalities this paper deals with state constrained optimal control problems governed by semilinear elliptic equations or variational inequalities by using ekeland s principle a minimum principle of pontryagin s type under some stability conditions of the optimal cost with respect to the state constraints is derived

optimal control for quasi newtonian flows with defective boundary conditions we consider a generalized newtonian fluid with defective boundary conditions where only flow rates or mean pressures are prescribed on parts of boundary the defect boundary condition problem is formulated as an optimal control problem in which a neumann or dirichlet boundary control is used for matching given flow rates or mean pressures for the constrained optimization problem an optimality system is derived from which a solution of the problem is obtained computational algorithms are discussed and numerical results are also presented

extremum entrop y based heisenberg like uncertainty relations in this work we use the extremization method of various information theoretic measures fisher information shannon entropy tsallis entropy for d dimensional quantum systems which complementarily describe the spreading of the quantum states of natural systems under some given constraints usually one or two radial expectation values this variational method allows us to determine an extremum entropy distribution which is the least biased one to characterize the state among all those compatible with the known data then we use it together with the spin dependent uncertainty like relations of daubechies thakkar type as a tool to obtain relationships between the position and momentum radial expectation values of the type for d dimensional systems of n fermions with spin s the resulting uncertainty like products which take into account both spatial and spin degrees of freedom of the fermionic constituents of the system are shown to often improve the best corresponding relationships existing in the literature

hardy lieb thirring inequalities for fractional schrodinger operators we show that the lieb thirring inequalities on moments of negative eigenvalues of schrodinger like operators remain true with possibly different constants when the critical hardy weight c x 2 is subtracted from the laplace operator we do so by first establishing a sobolev inequality for such operators similar results are true for fractional powers of the laplacian and the hardy weight and in particular for relativistic schrodinger operators we also allow for the inclusion of magnetic vector potentials as an application we extend for the first time the proof of stability of relativistic matter with magnetic fields all the way up to the critical value of the nuclear charge z 2 for less than some critical value

numerical solutions of the thermistor equations abstract this paper examines heat conduction in a thermistor used as a current surge regulator the problem consists of coupled nonlinear nonlocal parabolic initial boundary value problems simplifying assumptions are made which lead to two different problems each of which consists of a one space dimensional nonlocal parabolic initial boundary value problem numerical methods for the approximate solutions of both the steady state and the transient problems are described and the results of the numerical experiments are presented

analysis of neumann boundary optimal control problems for the stationary boussinesq equations including solid media this article deals with neumann boundary optimal control problems associated with the boussinesq equations including solid media these problems are first put into an appropriate mathematical formulation then the existence of optimal solutions is proved the use of lagrange multiplier techniques is justified and an optimality system of equations is derived

non validity of index law in fractional calculus a fractional differential operator with markovian and non markovian properties abstract we presented an analysis of evolutions equations generated by three fractional derivatives namely the riemann liouville caputo fabrizio and the atangana baleanu fractional derivatives for each evolution equation we presented the exact solution for time variable and studied the semigroup principle the riemann liouville fractional operator verifies the semigroup principle but the associate evolution equation does not the caputo fabrizio fractional derivative does not satisfy the semigroup principle but surprisingly the exact solution satisfies very well all the principle of semigroup however the atangana baleanu for small time is the stretched exponential derivative which does not satisfy the semigroup as operators for a large time the atangana baleanu derivative is the same with riemann liouville fractional derivative thus satisfies semigroup principle as an operator the solution of the associated evolution equation does not satisfy the semigroup principle as riemann liouville with the connection between semigroup theory and the markovian processes we found out that the atangana baleanu fractional derivative has at the same time markovian and non markovian processes we concluded that the fractional differential operator does not need to satisfy the semigroup properties as they portray the memory effects which are not always markovian we presented the exact solutions of some evolutions equation using the laplace transform in addition to this we presented the numerical solution of a nonlinear equation and show that the model with the atangana baleanu fractional derivative has random walk for small time we also observed that the mittag leffler function is a good filter than the exponential and power law functions which makes the atangana baleanu fractional derivatives powerful mathematical tools to model complex real world problems

aw1 p estimate for solutions to mixed boundary value problems for second order elliptic differential equations on demontre que sous des hypotheses plutot faibles une solution d un probleme aux valeurs limites mixte pour une equation elliptique d ordre 2 est dans l espace de sobolev w 1 p pour p 2

optimal control for the thermistor problem this paper is concerned with the state constrained optimal control of the two dimensional thermistor problem a quasi linear coupled system of a parabolic and elliptic pde with mixed boundary conditions this system models the heating of a conducting material by means of direct current existence uniqueness and continuity for the state system are derived by employing maximal elliptic and parabolic regularity by similar arguments the linearized state system is discussed while the adjoint system involving measures is investigated using a duality argument these results allow us to derive first order necessary conditions for the optimal control problem

a survey on fuzzy fractional differential and optimal control nonlocal evolution equations abstract we survey some representative results on fuzzy fractional differential equations controllability approximate controllability optimal control and optimal feedback control for several different kinds of fractional evolution equations optimality and relaxation of multiple control problems described by nonlinear fractional differential equations with nonlocal control conditions in banach spaces are considered

on a decoupled algorithm for solving a finite element problem for the approximation of viscoelastic fluid flow the aim of this work is to study a decoupled algorithm of a fixed point for solving a finite element fe problem for the approximation of viscoelastic fluid flow obeying an oldroyd b differential model the interest for this algorithm lies in its applications to numerical simulation and in the cost of computing furthermore it is easy to bring this algorithm into play the unknowns are sigma the viscoelastic part of the extra stress tensor u the velocity and p the pressure we suppose that the solution sigma u p is sufficiently smooth and small the approximation of stress velocity and pressure are resp p 1 discontinuous p 2 continuous p 1 continuous fe upwinding needed for convection of sigma is made by discontinuous fe the method consists to solve alternatively a transport equation for the stress and a stokes like problem for velocity and pressure previously results of existence of the solution for the approximate problem and error bounds have been obtained using fixed point techniques with coupled algorithm in this paper we show that the mapping of the decoupled fixed point algorithm is locally in a neighbourhood of sigma u p contracting and we obtain existence unicity locally of the solution of the approximate problem and error bounds

levy lieb constrained search formulation as a minimization of the correlation functional the constrained search formulation of levy and lieb which formally defines the exact hohenberg kohn functional for any n representable electron density is here shown to be equivalent to the minimization of the correlation functional with respect to the n 1 conditional probability density where n is the number of electrons of the system the consequences and implications of such a result are here analysed and discussed via a practical example

numerical analysis of a three fields model for a quasi newtonian flow abstract we study a finite element approximation of a non linear three fields version of stokes system for incompressible fluids such a problem arises when one considers a white metzner type model for a viscoelastic fluid flow and sets the relaxation time to zero the non linear aspect comes from the fact that a fraction of the viscosity of the fluid depends on the second invariant of the rate of deformation tensor we give existence and unicity results for the continuous problem and its approximation and we prove the error bound

optimal control problem for stochastic evolution equations in hilbert spaces in this article we consider an optimal control problem in which the controlled state dynamics is governed by a stochastic evolution equation in hilbert spaces and the cost functional has a quadratic growth the existence and uniqueness of the optimal control are obtained by the means of an associated backward stochastic differential equations with a quadratic growth and an unbounded terminal value as an application an optimal control of stochastic partial differential equations with dynamical boundary conditions is also given to illustrate our results

boundary control of semilinear elliptic equations with pointwise state constraints this paper is concerned with state constrained optimal control problems of semilinear elliptic equations the control being on the boundary optimality conditions are derived and regularity of the optimal solution is investigated

functionals of gegenbauer polynomials and d dimensional hydrogenic momentum expectation values the system of gegenbauer or ultraspherical polynomials cn x n 0 1 is a classical family of polynomials orthogonal with respect to the weight function x 1 x2 1 2 on the support interval 1 1 integral functionals of gegenbauer polynomials with integrand f x cn x 2 x where f x is an arbitrary function which does not depend on n or are considered in this paper first a general recursion formula for these functionals is obtained then the explicit expression for some specific functionals of this type is found in a closed and compact form namely for the functionals with f x equal to 1 x 1 x log 1 x2 and 1 x log 1 x which appear in numerous physico mathematical problems finally these functionals are used in the explicit evaluation of the momentum expectation values p and log p of the d dimensional hydrogenic atom with nuclear charge z 1 the power expectation values p are given by means of a terminating 5f4 hypergeometric function with unit argument which is