Linear functionals on nonlinear spaces and applications to problems from viscoplasticity theory

A classical result in the theory of monotone operators states that if C is a reflexive Banach space, and an operator A: C→C^* is monotone, semicontinuous and coercive, then A is surjective. In this paper, we define the ‘dual space’ C^* of a convex, usually not linear, subset C of some Banach space X (in general, we will have C^*⊃X^*) and prove an analogous result. Then, we give an application to problems from viscoplasticity theory, where the natural space to look for solutions is not linear. Copyright © 2007 John Wiley & Sons, Ltd.     

An approach to nonlinear programming

A nonlinear programming problem with inequality constraints and with unknown vectorx is converted to an unconstrained minimization problem in unknownsx and , where is a vector of Lagrange multipliers. It is shown that, if the original problem possesses standard convexity properties, then local minima of the associated unconstrained problem are in fact global minima of that problem and, consequently, Kuhn-Tucker points for the original problem. A computational procedure based on the conjugate residual scheme is applied in thex-space to solve the associated unconstrained problem. The resulting algorithm requires only first-order derivative information on the functions involved and will solve a quadratic programming problem in a finite number of steps.This research was supported by the National Science Foundation under Grant No. GK-16125. The author is indebted to the referee for several valuable comments and suggestions for improvement.     

An alternative approach to nonlinear filtering

The structure of a nonlinear filter with observation process having continuous and discontinuous components is considered. The approach is based on the so-called “Bayes” formula for conditional expectations. “Fubini” type theorems for stochastic integrals are given and used to obtain the representations of an optimal estimate and of the conditional likelihood ratio. A linear unnormalized filtering equation for controlled system process is derived.     

Deterministic homogenization of integral functionals with convex integrands

In order to widen the scope of the applications of deterministic homogenization, we consider here the homogenization problem for a family of integral functionals. The homogenization procedure tending to be classical, the choice focused on the convex integral functionals is made just to simplify the presentation of the paper. We use a new approach based on the Stepanov type spaces, which approach allows us to solve various problems such as the almost periodic homogenization problem and others without resorting to additional assumptions. We then apply it to obtain a general homogenization result and then we provide a number of physical applications of the result. The convergence method used falls within the scope of two-scale convergence.     

Scalar auxiliary variable approach for conservative/dissipative partial differential equations with unbounded energy functionals

BIT Numerical Mathematics - In this paper, we present a novel investigation of the so-called SAV approach, which is a framework to construct linearly implicit geometric numerical integrators for...     

An approach via symmetrization methods to nonlinear elliptic problems with a lower order term

In this paper we consider a class of nonlinear elliptic problems of the type

Fourier-Feynman transforms of unbounded functionals on abstract Wiener space

Huffman, Park and Skoug established several results involving Fourier-Feynman transform and convolution for functionals in a Banach algebra S on the classical Wiener space. Chang, Kim and Yoo extended these results to abstract Wiener space for a more generalized Fresnel class $ \mathcal{F}_{\mathcal{A}_1 ,\mathcal{A}_2 } $ \mathcal{F}_{\mathcal{A}_1 ,\mathcal{A}_2 } _A1,A2 than the Fresnel class $ \mathcal{F} $ \mathcal{F} (B)which corresponds to the Banach algebra S. In this paper we study Fourier-Feynman transform, convolution and first variation of unbounded functionals on abstract Wiener space having the form

Attractors for nonlinear reaction-diffusion systems in unbounded domains via the method of short trajectories

We consider a nonlinear reaction-diffusion equation on the whole space Rd. We prove the well-posedness of the corresponding Cauchy problem in a general functional setting, namely, when the initial datum is uniformly locally bounded in L² only. Then we adapt the short trajectory method to establish the existence of the global attractor and, if d?3, we find an upper bound of its Kolmogorov's ε-entropy.     

An abstract approach to nonlinear boltzmann-type equations

Summary Using a general existence and uniqueness theory for linear time dependent kinetic equations, for general inhomogeneous multidimensional spatial and velocity domains and partially absorbing boundaries, we obtain local in time solutions of a class of nonlinear Boltzmann type equations. For small initial-boundary data we obtain global in time solutions. The ideal norm on certain ideals in the Banach space ofL _p-functions on phase space is used to measure the ?size? of initial-boundary data and solutions. Kaniel-Shinbrot type upper and lower approximation arguments are applied. The combined length of the time interval of existence when applying the method repeatedly is analyzed as a function of the size of the initial-boundary data. Specific applications to the nonlinear Boltzmann equation itself and to the plane Broadwell model are given. Research conducted under the auspices of C.N.R. (Consiglio Nazionale delle Ricerche), Gruppo Fisica-Matematica, and partially supported by M.P.I. (Ministero della Pubblica Intruzione). Research conducted as a visiting professor supported by C.N.R., Gruppo Fisica-Matematica. Permanente address: Dept. of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.     

A BSDE approach to Nash equilibrium payoffs for stochastic differential games with nonlinear cost functionals

In this paper, we study Nash equilibrium payoffs for two-player nonzero-sum stochastic differential games via the theory of backward stochastic differential equations. We obtain an existence theorem and a characterization theorem of Nash equilibrium payoffs for two-player nonzero-sum stochastic differential games with nonlinear cost functionals defined with the help of doubly controlled backward stochastic differential equations. Our results extend former ones by Buckdahn et al. (2004) [3] and are based on a backward stochastic differential equation approach.     

A homogenization approach to flashing ratchets

The paper continues the program of the authors to develop a mathematical framework to understand and characterize the notion of “asymmetric” potentials, which has been introduced to explain how molecular motors work, considering flashing ratchets, i.e., molecules diffusing in a potential with periodic switches. The mathematical model is a Fokker–Planck equation with a space–time periodic potential and diffusion of order of magnitude compatible with the period of the potential. After performing a homogenization analysis of the problem the “asymmetric” potentials are characterized by the property that the solution, which models the molecule density, concentrates on one end of the domain. Finally explicit examples are presented exhibiting that the concentration phenomena (motor effect) takes place are presented. The proof uses techniques from the theory of viscosity solutions for the Hamilton–Jacobi equation which, in the homogenization limit, defines the effective hamiltonian.     

An approach to symmetrization via polarization

We prove that the Steiner symmetrization of a function can be approximated in by a sequence of very simple rearrangements which are called polarizations. This result is exploited to develop elementary proofs of many inequalities, including the isoperimetric inequality in Euclidean space. In this way we also obtain new symmetry results for solutions of some variational problems. Furthermore we compare the solutions of two boundary value problems, one of them having a "polarized" geometry and we show some pointwise inequalities between the solutions. This leads to new proofs of well-known functional inequalities which compare the solutions of two elliptic or parabolic problems, one of them having a "Steiner-symmetrized" geometry. The method also allows us to investigate the case of equality in the inequalities. Roughly speaking we prove that the equality sign is valid only if the original problem has the symmetrized geometry.

     

Approximation of nonlinear integral functionals

Doklady Mathematics -     

An alternative approach to the structure theory of PI-rings

We expose a rather simple and direct approach to the structure theory of prime PI-rings (“Posner’s theorem”), based on fundamental properties of the extended centroid of a prime ring.