a numerical method for solving a nonlinear 2 d optimal control problem with the classical diffusion equation this paper presents a numerical solution for solving a nonlinear 2 d optimal control problem 2dop the performance index of a nonlinear 2dop is described with a state and a control function furthermore dynamic constraint of the system is given by a classical diffusion equation it is preferred to use the ritz method for finding the numerical solution of the problem the method is based upon the legendre polynomial basis by using this method the given optimisation nonlinear 2dop reduces to the problem of solving a system of algebraic equations the benefit of the method is that it provides greater flexibility in which the given initial and boundary conditions of the problem are imposed moreover compared with the eigenfunction method the satisfactory results are obtained only in a small number of polynomials order this numerical approach is applicable and effective for such a kind of nonlinear 2dop the convergence of the method is extensively discussed and finally two illustrative example

the total variation flow in rn the purpose of this chapter is to prove existence and uniqueness of the minimizing total variation flow in n frac partial u partial t div left frac du left du right right in left 0 infty right times mathbb r n 3 1 coupled with the initial condition u 0 x u 0 x text x in mathbb r n 3 2 when u0 l loc 1 l n

a kohn sham system at zero temperature a one dimensional kohn sham system for spin particles is considered which effectively describes semiconductor nanostructures and which is investigated at zero temperature we prove the existence of solutions and derive a priori estimates for this purpose we find estimates for eigenvalues of the schrodinger operator with effective kohn sham potential and obtain w1 2 bounds of the associated particle density operator afterwards compactness and continuity results allow us to apply schauder s fixed point theorem in the case of vanishing exchange correlation potential uniqueness is shown by monotonicity arguments finally we investigate the behavior of the system if the temperature approaches zero

qualitative behavior and traveling waves for flux saturated porous media equations arising in optimal mass transportation abstract this paper is devoted to the analysis of qualitative properties of porous media flux saturated type operators which combine the effects of reaction terms of fkpp type in dimension one specifically we study the existence of traveling wave profiles sub and super solutions waiting time and dynamical properties of the support the paper also deals with the connection between these minimal traveling waves with those of other related models porous media flux saturated mechanism and the heat equation

on the relativistic heat equation in one space dimension we study the relativistic heat equation in one space dimension we prove a local regularity result when the initial datum is locally lipschitz in its support we propose a numerical scheme that captures the known features of the solutions and allows for analysing further properties of their qualitative behaviour

equations with singular diffusivity recently models of faceted crystal growth and of grain boundaries were proposed based on the gradient system with nondifferentiable energy in this article we study their most basic forms given by the equations ut ux ux x and ut 1 a aux ux x where both of the related energies include a ux term of power one which is nondifferentiable at ux 0 the first equation is spatially homogeneous while the second one is spatially inhomogeneous when a depends on x these equations naturally express nonlocal interactions through their singular diffusivities infinitely large diffusion constant which make the profiles of the solutions completely flat the mathematical basis for justifying and analyzing these equations is explained and theoretical and numerical approaches show how the solutions of the equations evolve

a strongly degenerate quasilinear equation the parabolic case we prove the existence and uniqueness of entropy solutions of the neumann problem for the quasilinear parabolic equation ut a u du where a z f z and f is a convex function of with linear growth as satisfying other additional assumptions in particular this class includes the case where f z z 0 and is a convex function with linear growth as

asymptotic estimates on uncertainty of the elastic moduli of completely random trigonal polycrystals because of the uncertain irregular microgeometry a random polycrystalline aggregate the shape and crystalline orientations of the constituent anisotropic grains of which are uncorrelated may have elastic moduli scattered over some though small intervals while the conventional macroscopic homogeneity and isotropy hypotheses for it may be considered only as approximate with corresponding uncertainty our formal bounds are proposed to provide the estimates on those uncertainties in asymptotic sense explicit expressions for the aggregates of trigonal crystals classes 3m 32 3m are derived

on the entropy conditions for some flux limited diffusion equations abstract in this paper we give a characterization of the notion of entropy solutions of some flux limited diffusion equations for which we can prove that the solution is a function of bounded variation in space and time this includes the case of the so called relativistic heat equation and some generalizations for them we prove that the jump set consists of fronts that propagate at the speed given by rankine hugoniot condition and we give on it a geometric characterization of the entropy conditions since entropy solutions are functions of bounded variation in space once the initial condition is to complete the program we study the time regularity of solutions of the relativistic heat equation under some conditions on the initial datum an analogous result holds for some other related equations without additional assumptions on the initial condition

variational estimates for the creep behaviour of polycrystals a variational procedure is developed for estimating the effective constitutive behaviour of polycrystalline materials undergoing high temperature creep the procedure is based on a new variational principle allowing the determination of the effective potential function of a given nonlinear polycrystal in terms of the corre sponding potential for a linear comparison polycrystal with an identical geometric arrangements of its constituent single crystal grains as such it constitutes an extension to locally anisotropic behaviour of the variational procedure devel oped by ponte castaneda 1991 for nonlinear heterogeneous media with locally isotropic behaviour by way of an example the procedure is applied to the de termination of bounds of the hashin shtrikman type for the effective potentials of statistically isotropic nonlinear polycrystals the bounds are computed for the special class of untextured fcc polycrystals with isotropic pure power law viscous behaviour first considered by hutchinson 1976 in the context of a calculation of the self consistent type the new bounds are found to be more restrictive than the corresponding classical taylor bishop hill bounds and also more re strictive if only slightly so than related bounds of the hashin shtrikman type by dendievel et al 1991 the new procedure has the advantage over the self consistent procedure of hutchinson 1976 that it may be applied without any essential complications to aggregates of crystals with slip systems exhibiting dif ferent creep rules with for example different power exponents and to general loading conditions however the distinctive feature of the new variational proce dure is that it may be used in conjunction with other types of known bounds and estimates for linear polycrystals to generate corresponding bounds and estimates for nonlinear polycrystals

vibration and buckling of multilayered composite beams according to higher order deformation theories natural frequencies and buckling stresses of laminated composite beams are analyzed by taking into account the complete effects of transverse shear and normal stresses and rotatory inertia by using the method of power series expansion of displacement components a set of fundamental dynamic equations of a one dimensional higher order theory for laminated composite beams subjected to axial stress is derived through hamilton s principle several sets of truncated approximate theories are applied to solve the eigenvalue problems of a simply supported laminated composite beam in order to ensure the accuracy of the present theory convergence properties of the first seven natural frequencies are examined in detail numerical results are compared with those of the published existing theories and fem solutions the modal displacement and stress distributions in the depth direction are obtained and plotted in figures the present global higher order approximate theories can predict the natural frequencies buckling stresses and interlaminar stresses of multilayered composite beams as accurately as three dimensional elasticity solutions

existence and uniqueness for a degenerate parabolic equation with l1 data in this paper we study existence and uniqueness of solutions for the boundary value problem with initial datum in l1 ut div a x du in 0 u a u on 0 u x 0 u0 x in where a is a caratheodory function satisfying the classical leray lions hypothesis a is the neumann boundary operator associated to a du the gradient of u and is a maximal monotone graph in r r with 0 0

on a nonlinear flux limited equation arising in the transport of morphogens abstract motivated by a mathematical model for the transport of morphogens in biological systems we study existence and uniqueness of entropy solutions for a mixed initial boundary value problem associated with a nonlinear flux limited diffusion system from a mathematical point of view the problem behaves more as a hyperbolic system than a parabolic one

on optimal zeroth order bounds with application to hashin shtrikman bounds and anisotropy parameters abstract a new result presented in this paper is the evaluation of the hashin shtrikman bounds for composites composed of arbitrarily anisotropic constituents to date evaluation of the hashin shtrikman bounds are limited to composites with isotropic constituents or to polycrystalline composites with specific crystal symmetries the generality of the exact result presented herein is achieved through a reinterpretation of kroner s j mech phys solids 25 1977 137 recursive relations for n th order bounds and the optimal zeroth order n 0 bound the definitions of optimal zeroth order bounds are extended to all even ordered tensors and procedures are presented to evaluate these bounds for all second and fourth order tensors while optimal zeroth order bounds are not new the ability to calculate them for fourth order tensors of arbitrary symmetry is new utilizing the zeroth order bounds material anisotropy parameters are defined which quantify the extent of anisotropy for even ordered tensors

finite propagation speed for limited flux diffusion equations we prove that the support of solutions of a limited flux diffusion equation known as a relativistic heat equation evolves at a constant speed identified as the speed of light c for that we construct entropy sub and super solutions which are fronts evolving at speed c and prove the corresponding comparison principle between entropy solutions and sub and super solutions respectively this enables us to prove the existence of discontinuity fronts moving at light s speed

hyperbolic phenomena in a strongly degenerate parabolic equation we consider the equation u t u u x x where 0 and where is a strictly increasing function with lim s we solve the associated cauchy problem for an increasing initial function and discuss to what extent the solution behaves qualitatively like solutions of the first order conservation law u t u x equations of this type arise for example in the theory of phase transitions where the corresponding free energy functional has a linear growth rate with respect to the gradient

diffusion front capturing schemes for a class of fokker planck equations application to the relativistic heat equation in this research work we introduce and analyze an explicit conservative finite difference scheme to approximate the solution of initial boundary value problems for a class of limited diffusion fokker planck equations under homogeneous neumann boundary conditions we show stability and positivity preserving property under a courant friedrichs lewy parabolic time step restriction we focus on the relativistic heat equation as a model problem of the mentioned limited diffusion fokker planck equations we analyze its dynamics and observe the presence of a singular flux and an implicit combination of nonlinear effects that include anisotropic diffusion and hyperbolic transport we present numerical approximations of the solution of the relativistic heat equation for a set of examples in one and two dimensions including continuous initial data that develops jump discontinuities in finite time we perform the numerical experiments through a class of explicit high order accurate conservative and stable numerical schemes and a semi implicit nonlinear crank nicolson type scheme

vibration and buckling of deep beam columns on two parameter elastic foundations natural frequencies and buckling stresses of a deep beam column on two parameter elastic foundations are analyzed by taking into account the effect of shear deformation depth change the transverse displacement w can vary in the depth direction of beam columns and rotatory inertia by using the method of power series expansion of displacement components a set of fundamental dynamic equations of a one dimensional higher order theory for thin rectangular beam columns subjected to axial stress is derived through hamilton s principle several sets of truncated approximate theories are applied to solve the eigenvalue problems of a simply supported deep elastic beam column in order to assure the accuracy of the present theory convergence properties of the minimum natural frequency and the buckling stress are examined in detail it is noted that the present approximate theories can predict the natural frequencies and buckling stress of deep beam columns on elastic foundations accurately compared with the timoshenko beam theory and the classical beam theory

a fisher kolmogorov equation with finite speed of propagation abstract in this paper we study a fisher kolmogorov type equation with a flux limited diffusion term and we prove the existence and uniqueness of finite speed moving fronts and the existence of some explicit solutions in a particular regime of the equation

uniqueness of renormalized solutions of degenerate elliptic parabolic problems we consider a general class of degenerate elliptic paraboic problems associated with the equation b v t div a v dv f using kruzhkov s method of doubling variables both in space and time we prove uniqueness and a comparison principle in l 1 for renormalized solutions

the dirichlet problem associated to the relativistic heat equation we prove existence and uniqueness of entropy solutions for the nonhomogeneous dirichlet problem associated to the relativistic heat equation

some regularity results on the relativistic heat equation we prove some partial regularity results for the entropy solution u of the so called relativistic heat equation in particular under some assumptions on the initial condition u0 we prove that ut t is a radon measure in rn moreover if u0 is log concave inside its support being a convex set then we show the solution u t is also log concave in its support t this implies its smoothness in t in that case we can give a simpler characterization of the notion of entropy solution

anisotropic tempered diffusion equations abstract we introduce a functional framework which is specially suited to formulate several classes of anisotropic evolution equations of tempered diffusion type under an amenable set of hypothesis involving a very natural potential function these models can be shown to belong to the entropy solution framework devised by andreu et al 2005 therefore ensuring well posedness we connect the properties of this potential with those of the associated cost function thus providing a link with optimal transport theory and a supply of new examples of relativistic cost functions moreover we characterize the anisotropic spreading properties of these models and we determine the rankine hugoniot conditions that rule the temporal evolution of jump hypersurfaces under the given anisotropic flows

local in time regularity results for some flux limited diffusion equations of porous media type abstract we prove some regularity results for solutions of some flux limited diffusion equations of porous media type for lipschitz initial data or assuming a uniform gradient bound on some power of the data including the fast diffusion case in which the results are global in time we also develop the existence and uniqueness theory for solutions of the fast diffusion case which was not covered in the current literature

on a nonlinear hyperbolic equation with a bistable reaction term abstract nonlinear second order hyperbolic equations are gaining ground as models in many areas of application as extensions of parabolic reaction diffusion equations that might otherwise be used the theory of travelling wave solutions of such reaction diffusion equations is well established the present paper is concerned with its counterpart for the wider class of equations in the particular case that the reaction term is bistable conditions that are necessary and sufficient for the existence and uniqueness of these solutions are determined a combination of traditional ordinary differential equation techniques and an innovatory integral equation approach is employed

free vibration and stability of thin elastic beams subjected to axial forces abstract natural frequencies and buckling loads of a simply supported beam with small length to depth ratio and sufficiently thin rectangular cross sections subjected to initial axial tensile and or compressive forces are analyzed by using the method of power series expansion of displacement components a set of fundamental dynamic equations of a one dimentional higher order beam theory for thin rectangular beams is derived through hamilton s principle several sets of truncated approximate theories which can take into account the effects of both shear deformations with depth changes and rotary inertia are applied to solve the eigenvalue problem of a thin elastic beam the navier solution procedure is used to satisfy the boundary conditions of a simply supported thin rectangular beam in order to assure the accuracy of the present theory convergence properties of the minimum natural frequency and the buckling load for the axial and bending problems of thin beams are examined in detail it is noticed that the present approximate theories can predict the natural frequencies and buckling loads of thin beams with small length to depth ratio more accurately than other refined higher order theories and the classical beam theory

a strongly degenerate quasilinear elliptic equation abstract we prove existence and uniqueness of entropy solutions for the quasilinear elliptic equation u div a u du v where 0 v l 1 r n l r n a z f z and f is a convex function of with linear growth as satisfying other additional assumptions in particular this class of equations includes the elliptic problems associated to a relativistic heat equation and a flux limited diffusion equation used in the theory of radiation hydrodynamics respectively in a second part of this work using crandall liggett s iteration scheme this result will permit us to prove existence and uniqueness of entropy solutions for the corresponding parabolic cauchy problem

existence and uniqueness of a solution for a parabolic quasilinear problem for linear growth functionals with l 1 data we introduce a new concept of solution for the dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth using kruzhkov s method of doubling variables both in space and time we prove uniqueness and a comparison principle in l 1 for these solutions to prove the existence we use the nonlinear semigroup theory

estimation of overall properties of random polycrystals with the use of invariant decompositions of hooke s tensor in the paper the theoretical analysis of bounds and self consistent estimates of overall properties of linear random polycrystals composed of arbitrarily anisotropic grains is presented in the study two invariant decompositions of hooke s tensors are used the applied method enables derivation of novel expressions for estimates of the bulk and shear moduli which depend on invariants of local stiffness tensor with use of these expressions the materials are considered for which at the local level constraints are imposed on deformation or some stresses are unsustained