Search results for "Mathematical analysis"
showing 10 items of 2409 documents
Spaces of Operator-valued Functions Measurable with Respect to the Strong Operator Topology
2009
Let X and Y be Banach spaces and (Ω, Σ, μ) a finite measure space. In this note we introduce the space L p /μ; ℒ(X, Y)] consisting of all (equivalence classes of) functions Φ:Ω↦ℒ(X, Y) such that ω↦Φ(ω)x is strongly μ-measurable for all x∈X and ω↦Φ(ω)f(ω) belongs to L 1(μ; Y) for all f∈L p′ (μ; X), 1/p+1/p′=1. We show that functions in L p /μ; ℒ(X, Y)] define operator-valued measures with bounded p-variation and use these spaces to obtain an isometric characterization of the space of all ℒ(X, Y)-valued multipliers acting boundedly from L p (μ; X) into L q (μ; Y), 1≤q<p<∞.
Well-posedness of the boundary layer equations
2004
We consider the mild solutions of the Prandtl equations on the half space. Requiring analyticity only with respect to the tangential variable, we prove the short time existence and the uniqueness of the solution in the proper function space. Theproof is achieved applying the abstract Cauchy--Kowalewski theorem to the boundary layer equations once the convection-diffusion operator is explicitly inverted. This improves the result of [M. Sammartino and R. E. Caflisch, Comm. Math. Phys., 192 (1998), pp. 433--461], as we do not require analyticity of the data with respect to the normal variable.
Existence and stability of periodic solutions in a neural field equation
2017
We study the existence and linear stability of stationary periodic solutions to a neural field model, an intergo-differential equation of the Hammerstein type. Under the assumption that the activation function is a discontinuous step function and the kernel is decaying sufficiently fast, we formulate necessary and sufficient conditions for the existence of a special class of solutions that we call 1-bump periodic solutions. We then analyze the stability of these solutions by studying the spectrum of the Frechet derivative of the corresponding Hammerstein operator. We prove that the spectrum of this operator agrees up to zero with the spectrum of a block Laurent operator. We show that the no…
Some results about operators in nested Hilbert spaces
2005
With the use of interpolation methods we obtain some results about the domain of an operator acting on the nested Hilbert space {ℋf}f∈∑ generated by a self-adjoint operatorA and some estimates of the norms of its representatives. Some consequences in the particular case of the scale of Hilbert spaces are discussed.
Generalized modulational instability in multimode fibers: Wideband multimode parametric amplification
2015
In this paper intermodal modulational instability (IM-MI) is analyzed in a multimode fiber where several spatial and polarization modes propagate. The coupled nonlinear Schr\"odinger equations describing the modal evolution in the fiber are linearized and reduced to an eigenvalue problem. As a result, the amplification of each mode can be described by means of the eigenvalues and eigenvectors of a matrix that stores the information about the dispersion properties of the modes and the modal power distribution of the pump. Some useful analytical formulas are also provided that estimate the modal amplification as function of the system parameters. Finally, the impact of third-order dispersion …
Wave Propagation in a 3-D Optical Waveguide
2003
In this paper we study the problem of wave propagation in a 3-D optical fiber. The goal is to obtain a solution for the time-harmonic field caused by a source in a cylindrically symmetric waveguide. The geometry of the problem, corresponding to an open waveguide, makes the problem challenging. To solve it, we construct a transform theory which is a nontrivial generalization of a method for solving a 2-D version of this problem given by Magnanini and Santosa.\cite{MS} The extension to 3-D is made complicated by the fact that the resulting eigenvalue problem defining the transform kernel is singular both at the origin and at infinity. The singularities require the investigation of the behavio…
Existence for shape optimization problems in arbitrary dimension
2002
We discuss some existence results for optimal design problems governed by second order elliptic equations with the homogeneous Neumann boundary conditions or with the interior transmission conditions. We show that our continuity hypotheses for the unknown boundaries yield the compactness of the associated characteristic functions, which, in turn, guarantees convergence of any minimizing sequences for the first problem. In the second case, weaker assumptions of measurability type are shown to be sufficient for the existence of the optimal material distribution. We impose no restriction on the dimension of the underlying Euclidean space.
Optimization of continuous elastic perfectly plastic beams
2004
Abstract The optimal design of elastic plastic beams subjected to loads quasi-statically variable within a given domain is studied. A minimum volume formulation and a maximum load multiplier formulation are developed, both expressed on the grounds of a statical as well as of a kinematical approach. The search problems are formulated according to three different limiting criteria acting simultaneously. For each criterion a corresponding selected safety factor is imposed. The Euler–Lagrange equations related to the above described problems are deduced. A special iterative technique devoted to the solution of the above referenced problems is proposed. Some numerical examples conclude the paper.
Fixed domain approaches in shape optimization problems with Dirichlet boundary conditions
2009
Fixed domain methods have well-known advantages in the solution of variable domain problems including inverse interface problems. This paper examines two new control approaches to optimal design problems governed by general elliptic boundary value problems with Dirichlet boundary conditions. Numerical experiments are also included peerReviewed
Optimal Bounds on Plastic Deformations for Bodies Constituted of Temperature-Dependent Elastic Hardening Material
1997
Bounds are investigated on the plastic deformations in a continuous solid body produced during the transient phase by cyclic loading not exceeding the shakedown limit. The constitutive model employs internal variables to describe temperature-dependent elastic-plastic material response with hardening. A deformation bounding theorem is proved. Bounds turn out to depend on some fictitious self-stresses and mechanical internal variables evaluated in the whole structure. An optimization problem, aimed to make the bound most stringent, is formulated. The Euler-Lagrange equations related to this last problem are deduced and they show that the relevant optimal bound has a local character, i.e., it …