Showing papers in "Analysis and Applications in 2006"
TL;DR: In this paper, the authors characterize the reproducing kernel Hilbert spaces whose elements are p-integrable functions in terms of the boundedness of the integral operator whose kernel is the Reproducing Kernel.
Abstract: We characterize the reproducing kernel Hilbert spaces whose elements are p-integrable functions in terms of the boundedness of the integral operator whose kernel is the reproducing kernel. Moreover, for p = 2, we show that the spectral decomposition of this integral operator gives a complete description of the reproducing kernel, extending the Mercer theorem.
229 citations
TL;DR: In this paper, it was shown that for an open set Ω with sufficiently smooth boundary and homogeneous Dirichlet or Neumann limit conditions, the global and interior error is of order e.
Abstract: In a previous paper about homogenization of the classical problem of diffusion in a bounded domain with sufficiently smooth boundary, we proved that the global error is of order e1/2. Now, for an open set Ω with sufficiently smooth boundary and homogeneous Dirichlet or Neumann limit conditions, we show that in any open set strongly included in Ω the error is of order e. If the open set Ω ⊂ ℝn is of polygonal (n = 2) or polyhedral (n = 3) boundary, we also give the global and interior error estimates.
91 citations
TL;DR: In this article, a function space in which the Cauchy problem for the Boltzmann equation is well-posed globally in time near an absolute Maxwellian in a mild sense without any regularity conditions is presented.
Abstract: We present a function space in which the Cauchy problem for the Boltzmann equation is well-posed globally in time near an absolute Maxwellian in a mild sense without any regularity conditions. The asymptotic stability of the absolute Maxwellian is also established in this space and, moreover, it is shown that the higher order spatial derivatives of the solutions vanish in time faster than the lower order derivatives. No smallness assumptions are imposed on the derivatives of the initial data, and the optimal decay rates are derived. Furthermore, the Boltzmann equation with a time-periodic source term is solved in the same space on the unique existence and stability of a time-periodic solution which has the same period as the source term. The proof is based on the spectral analysis of the linearized Boltzmann operator.
66 citations
TL;DR: In this paper, the discretization of inverse problems defined by a Carleman operator is studied and a convergence analysis is given for the class of inverse Carlemen problems.
Abstract: We study the discretization of inverse problems defined by a Carleman operator. In particular, we develop a discretization strategy for this class of inverse problems and we give a convergence analysis. Learning from examples, as well as the discretization of integral equations, can be analyzed in our setting.
35 citations
TL;DR: The authors derived nonlinearly elastic membrane plate models for hyperelastic incompressible materials using Γ-convergence arguments and obtained an integral representation of the limit two-dimensional internal energy owing to a result of singular functionals relaxation due to Ben Belgacem.
Abstract: In this paper, we derive nonlinearly elastic membrane plate models for hyperelastic incompressible materials using Γ-convergence arguments. We obtain an integral representation of the limit two-dimensional internal energy owing to a result of singular functionals relaxation due to Ben Belgacem [6].
25 citations
TL;DR: In this paper, the authors established the global existence and stability of a multidimensional conic shock wave for three-dimensional steady supersonic flow past an infinite cone, where the flow is assumed to be hypersonic and described by a steady potential flow equation.
Abstract: In this paper, we establish the global existence and stability of a multidimensional conic shock wave for three-dimensional steady supersonic flow past an infinite cone. The flow is assumed to be hypersonic and described by a steady potential flow equation. Under an appropriate boundary condition on the curved cone, we show that a pointed shock attached at the vertex of the cone will exist globally in the whole space.
23 citations
TL;DR: In this article, the authors present the physical model and the mathematical formulation, which lead to an original locally hyperbolic conservation law, after some heuristic simplifications, the definition of a solution and some mathematical tools in order to resolve the problem are given.
Abstract: In this paper, we are interested in the mathematical analysis of a geological stratigraphic model, taking into account a limited weathering condition. Firstly, we present the physical model and the mathematical formulation, which lead to an original locally hyperbolic conservation law. Then, after some heuristic simplifications, the definition of a solution and some mathematical tools in order to resolve the problem are given. Within the framework of sedimentation processes, we illustrate in the 1-D case the changing type of the equation: hyperbolic with finite speed propagation and parabolic with diffusive comportment or stationary. At the end, we present some open problems related to free boundary phenomena.
20 citations
TL;DR: In this paper, the authors consider a nonlinear elliptic equation driven by the p-Laplacian with a discontinuous nonlinearity and prove the existence of at least two distinct solutions, both smooth and strictly positive.
Abstract: We consider a nonlinear elliptic equation driven by the p-Laplacian with a discontinuous nonlinearity. Such problems have a "multivalued" and a "single-valued" interpretation. We are interested in the latter and we prove the existence of at least two distinct solutions, both smooth and one strictly positive. Our approach is variational based on the nonsmooth critical point theory for locally Lipschitz functions, coupled with penalization and truncation techniques.
16 citations
TL;DR: In this paper, the authors prove existence and uniqueness results for the Oseen problem in exterior domains of ℝ3 and set the problem in weighted Sobolev spaces, which relies on a Lp-theory for any real p such that 1 < p < ∞.
Abstract: In this paper, we prove existence and uniqueness results for the Oseen problem in exterior domains of ℝ3. To prescribe the growth or decay of functions at infinity, we set the problem in weighted Sobolev spaces. The analysis relies on a Lp-theory for any real p such that 1 < p < ∞.
14 citations
TL;DR: In this paper, the uniqueness and non uniqueness of minimizers of functionals involving nonlocal quantities are studied, and it is shown that the uniqueness of a functional minimizer of the form ∫ a(∫Ω gu dx)|∇u|2 dx - 2 ∫ Δ − 2 √ fu dx is ensured if a > 0 and 1/a is strictly concave in the sense that (1/a) < 0 on (0, ∞).
Abstract: We study uniqueness and non uniqueness of minimizers of functionals involving nonlocal quantities. We give also conditions which lead to a lack of minimizers and we show how minimization on an infinite dimensional space reduces here to a minimization on ℝ. Among other things, we prove that uniqueness of minimizers of functionals of the form ∫Ω a(∫Ω gu dx)|∇u|2 dx - 2 ∫Ω fu dx is ensured if a > 0 and 1/a is strictly concave in the sense that (1/a)″ < 0 on (0, ∞).
11 citations
TL;DR: In this paper, the authors consider finite volume methods based on a non-uniform grid and prove convergence results for the finite volume function spaces and for some model elliptic and parabolic boundary value problems using these discretization spaces.
Abstract: In this article, we consider finite volume methods based on a non-uniform grid. Finite volume methods are compared to finite difference methods based on a related grid. As an application, various convergence results are proved for the finite volume function spaces and for some model elliptic and parabolic boundary value problems using these discretization spaces.
TL;DR: In this paper, it was shown that if the set Ω is bounded and has a smooth boundary, then the mapping C ↦ Θ is of class C∞ between manifolds in appropriate Banach spaces.
Abstract: A basic theorem from differential geometry asserts that if the Riemann curvature tensor associated with a smooth field C of positive-definite symmetric matrices of order n vanishes in a simply-connected open subset Ω of ℝn, then C is the metric tensor of a manifold isometrically immersed in ℝn. If Ω is connected, then the isometric immersion Θ defined in this fashion is unique up to isometries of ℝn. We prove that if the set Ω is bounded and has a smooth boundary, then the mapping C ↦ Θ is of class C∞ between manifolds in appropriate Banach spaces.
TL;DR: In this article, the boundary value problem for the stationary Stokes equations in an infinite sector with the slip and the stress boundary conditions is studied and the existence of the unique solution is proved in weighted Sobolev spaces by means of the Mellin transform.
Abstract: As a model problem of the stationary free boundary problem for the Navier–Stokes equations in a vessel whose wall has a contact with a free surface, we are concerned in this paper with the boundary value problem for the stationary Stokes equations in an infinite sector with the slip and the stress boundary conditions. Existence of the unique solution is proved in weighted Sobolev spaces by means of the Mellin transform.
TL;DR: In this paper, the authors considered the Cauchy problem for an O.D. with discontinuous right-hand side in an L∞ space and proved that the O.E. has a unique Caratheodory solution, which depends Lipschitz continuously on the data.
Abstract: We consider the Cauchy problem for an ordinary differential equation with discontinuous right-hand side in an L∞ space. Under the assumptions that the vector field is directionally continuous with bounded directional variation, we prove that the O.D.E. has a unique Caratheodory solution, which depends Lipschitz continuously on the data.
TL;DR: In this article, a new uniform asymptotic approximation for the late coefficients of the modal expansion of a plane wave from a circular dielectric cylinder is presented, which is valid for the entire region exterior to the cylinder, including the transition region.
Abstract: The classical modal expansion for the scattered field of a plane wave from a circular dielectric cylinder is studied. A new uniform asymptotic approximation is presented for the late coefficients in this expansion, in the case of a fixed relative dielectric constant er, both real and complex. These new approximations for the mode values are not based on the scattering matrix but rather the classical WKBJ approximations for the Bessel functions, and are valid for the entire region exterior to the cylinder, including the transition region. Furthermore, a precise asymptotic form for the location of a certain critical Regge pole is obtained. It is shown that this pole can lead to at least one dramatic resonant modal term at certain critical values, and the exponential nature of the mode in question is determined explicitly. This is followed by an extension to complex values of er with new uniform asymptotic approximations for the modes also being obtained, and these in turn demonstrate a heavy damping of the resonant mode.
TL;DR: In this article, a general property of the stability pattern of successive periodic solutions of the first-order scalar differential equation u′ = f(t, u) is given.
Abstract: Using a monotonicity property specific to this case, we give a general property of the stability pattern of successive periodic solutions of the first-order scalar differential equation u′ = f(t, u). We also give an optimal smallness condition in order for the quasi-autonomous equation u′+g(u) = f(t) where g ∈ C1(ℝ) and f : ℝ → ℝ is almost periodic to have exactly N almost periodic solutions on the line assuming that we have exactly N equilibria ci and g′(ci) ≠ 0 for all i.
TL;DR: In this paper, the authors considered the large n behavior of the hypergeometric polynomial 3F_2 and derived an integral representation of the 3F2 with respect to the Catalan-Larcombe-French sequence.
Abstract: The large n behavior of the hypergeometric polynomial \[ {}_3F_2 \Bigg(\begin{array}{c} -n, \frac{1}{2}, \frac{1}{2}\\ \frac{1}{2} -n,\frac{1}{2} -n \end{array};-1\Bigg) \] is considered by using integral representations of this polynomial. This 3F2 polynomial is associated with the Catalan–Larcombe–French sequence. Several other representations are mentioned, with references to the literature, and another asymptotic method is described by using a generating function of the sequence. The results are similar to those obtained by Clark (2004) who used a binomial sum for obtaining an asymptotic expansion.