The Helmholtz–Weyl decomposition in weighted Sobolev spaces

Let f∈L_{2, ? µ}(?³), where where x = (x₁, x₂, x₃) is the Cartesian system in ?³, x′ = (x₁, x₂), , µ∈?₊\?. We prove the decomposition f = ? ?u + g, with g divergence free and u is a solution to the problem in ?³ Given f∈L_{2, ? µ}(?³) we show the existence of u∈H(?³) such that where Since f, u, g are defined in ?³ we need a sufficiently fast decay of these functions as |x|→∞. Copyright © 2010 John Wiley & Sons, Ltd.     

Construction of vector valued wavelet packets on ℝ+ using Walsh-Fourier transform

In this paper, the concept of vector-valued wavelet packets in space L ²(?₊, ? ^N ) is introduced. Some properties of vector-valued wavelets packets are studied and orthogonality formulas of these wavelets packets are obtained. New orthonormal basis of L ²(?₊, ? ^N ) is obtained by constructing a series of subspaces of vector-valued wavelet packets.     

Irreducible ℤ<Subscript>+</Subscript>-modules of near-group fusion ring <Emphasis Type="Italic">K</Emphasis>(ℤ<Subscript>3</Subscript>, 3)

The near-group rings are an important class of fusion rings in the theory of tensor categories. In this paper, the irreducible ?₊-modules over the near-group fusion ring K(?₃, 3) are explicitly classified. It turns out that there are only four inequivalent irreducible ?₊-modules of rank 2 and two inequivalent irreducible ?₊-modules of rank 4 over K(?₃, 3).     

On the Robin Boundary Condition for Laplace's Equation in Lipschitz Domains

Abstract

Let Ω be a bounded Lipschitz domain in ? ⁿ , n ≥ 3 with connected boundary. We study the Robin boundary condition ?u/?N + bu = f ∈ L ^p (?Ω) on ?Ω for Laplace's equation Δu = 0 in Ω, where b is a non-negative function on ?Ω. For 1 < p < 2 + ?, under suitable compatibility conditions on b, we obtain existence and uniqueness results with non-tangential maximal function estimate ‖(?u)*‖ _p  ≤ C‖f‖ _p , as well as a pointwise estimate for the associated Robin function. Moreover, the solution u is represented by a single layer potential.     

Higher moments for kinetic equations: The Vlasov–Poisson and Fokker–Planck cases

Let us consider a solution f(x,v,t)?L¹(?^2N × [0,T]) of the kinetic equation where |v|^α+1 f_o,|v|^α ?L¹ (?2N × [0, T]) for some α< 0. We prove that f has a higher moment than what is expected. Namely, for any bounded set K_x, we have We use this result to improve the regularity of the local density ρ(x,t) = ∫?dν for the Vlasov–Poisson equation, which corresponds to g = E?, where E is the force field created by the repartition ? itself. We also apply this to the Bhatnagar-Gross-;Krook model with an external force, and we prove that the solution of the Fokker-Pianck equation with a source term in L² belongs to L²([0, T]; H^1/2(?)).     

Inequalities for the Gaussian hypergeometric function

we study the monotonicity of certain combinations of the Gaussian hypergeometric functions F(-1/2,1/2;1;1- xc) and F(-1/2- δ,1/2 + δ;1;1- xd) on(0,1) for given 0 c 5d/6 ∞ andδ∈(-1/2,1/2),and find the largest value δ1 = δ1(c,d) such that inequality F(-1/2,1/2;1;1- xc) F(-1/2- δ,1/2 + δ;1;1- xd) holds for all x ∈(0,1). Besides,we also consider the Gaussian hypergeometric functions F(a- 1- δ,1- a + δ;1;1- x3) and F(a- 1,1- a;1;1- x2) for given a ∈ [1/29,1) and δ∈(a- 1,a),and obtain the analogous results.     

Hamiltonian ODEs in the Wasserstein space of probability measures

In this paper we consider a Hamiltonian H on ??₂(?^2d), the set of probability measures with finite quadratic moments on the phase space ?^2d = ?^d × ?^d, which is a metric space when endowed with the Wasserstein distance W₂. We study the initial value problem dμ_t/dt + ? · (??_d v _tμ_t) = 0, where ??_d is the canonical symplectic matrix, μ₀ is prescribed, and v _t is a tangent vector to ??₂(?^2d) at μ_t, belonging to ?H(μ_t), the subdifferential of H at μ_t. Two methods for constructing solutions of the evolutive system are provided. The first one concerns only the case where μ₀ is absolutely continuous. It ensures that μ_t remains absolutely continuous and v _t = ?H(μ_t) is the element of minimal norm in ?H(μ_t). The second method handles any initial measure μ₀. If we further assume that H is λ‐convex, proper, and lower‐semicontinuous on ??₂(?^2d), we prove that the Hamiltonian is preserved along any solution of our evolutive system, H(μ_t) = H(μ₀). © 2007 Wiley Periodicals, Inc.     

An artificial boundary condition for an advection–diffusion equation

Consider the advection–diffusion equation: u₁ + au_x1 ? vδu = 0 in ?ⁿ × ?⁺ with initial data u⁰; the Support of u⁰ is contained in ?(x₁ < 0) and a: ?ⁿ → ? is positive. In order to approximate the full space solution by the solution of a problem in ? × ?⁺, we propose the artificial boundary condition: u₁ + au_x1 = 0 on ∑. We study this by means of a transmission problem: the error is an O(v²) for small values of the viscosity v.     

Index transforms associated with generalized hypergeometric functions

We deal with a class of integral transformations whose kernels contain the Clausenian hypergeometric function ₃F₂(a₁,a₂,a₃;b₁,b₂;z). These transforms are defined in terms of integrals with respect to their parameters. It involves as particular cases the familiar Olevskii and generalized Mehler–Fock transforms which are key tools in the methods of the mathematical theory of elasticity. The main theorem of boundedness of these operators as a map of L₂(?₊)L₂(?₊;x^?1 dx) is proved. Some examples of the Olevskii and Mehler–Fock type integrals are given. Copyright © 2003 John Wiley & Sons, Ltd.     

Schauder Estimates for the Single Layer Potential in Hydrodynamics

Let Ω be an open set in ?^N(N ? 3), with compact boundary ?Ω of type C^1,α(?(0,1)). We show that the single layer potential Ef, related to the stationary Stokes system on Ω, belongs to C^1,α(?Ω)^N, provided the source density f belongs to C^α(?Ω)^N. In addition, we prove a related estimate of the function E(f)_?Ω and its tangential derivatives.     

Small Data Scattering for the One-Dimensional Nonlinear Dirac Equation with Power Nonlinearity

We study scattering problems for the one-dimensional nonlinear Dirac equation (?_t + α?_x + iβ)Φ = λ|Φ|^p?1Φ. We prove that if p > 3 (resp. p > 3 + 1/6), then the wave operator (resp. the scattering operator) is well-defined on some 0-neighborhood of a weighted Sobolev space. In order to prove these results, we use linear operators D(t)xD(?t) and t?_x + x?_t ? α/2, where {D(t)}_t∈? is the free Dirac evolution group. For the reader's convenience, in an appendix we list and prove fundamental properties of D(t)xD(?t) and t?_x + x?_t ? α/2.     

An integral operator related to the Stokes system in exterior domains

Consider a bounded domain Ω in ?³ with C²-boundary ?Ω. In [1] the Stokes problem in the exterior domain ?³/Ω , with resolvent parameter [λ??\] ? [∞,0], is solved by using the method of integral equations. However, for estimating the corresponding solutions in L^p norms, it turns out that a certain operator defined on the spaces L^r(?Ω)³, for r ?]1, ∞[, has to be evaluated in the norm of L^r(?Ω)³. This estimate is proved in the present paper.     

Families of multiresolution and wavelet spaces with optimal properties

Under suitable conditions, if the scaling functions ?₁ and ?₂ generate the multiresolutions V _(j)(?₁) and V _(j)(?₂), then their convolution ?₁*?₂also generates a multiresolution V _(j)(?₁*?₂) More over, if p is an appropriate convolution operator from l ₂ into itself and if ? is a scaling function generating the multiresolution V _(j)(?),then p*?is a scaling function generating the same multiresolution V _(j)(?)=V _(j)(p*?). Using these two properties, we group the scaling and wavelet functions into equivalent classes and consider various equivalent basis functions of the associated function spaces We use the n-fold convolution product to construct sequences of multiresolution and wavelet spaces V _(j)(?ⁿ) and W _(j)(?ⁿ) with increasing regularity. We discuss the link between multiresolution analysis and Shannon's sampling theory. We then show that the interpolating and orthogonal pre- and post-filters associated with the multiresolution sequence V ₍₀₎(?ⁿ)asymptotically converge to the ideal lowpass filter of Shannon. We also prove that the filters associated with the sequence of wavelet spaces W ₍₀₎(?ⁿ)convergeto the ideal bandpass filter. Finally, we construct the basic wavelet sequences ψ ^b _nand show that they tend to Gabor functions. Thisprovides wavelets that are nearly time-frequency optimal. The theory is illustrated with the example of polynomial splines.     

Interpolation of operators of weak type (ϕ, ϕ)

Considering the measurable and nonnegative functions ? on the half-axis [0, ∞) such that ?(0) = 0 and ?(t) → ∞ as t → ∞, we study the operators of weak type (?, ?) that map the classes of ?-Lebesgue integrable functions to the space of Lebesgue measurable real functions on ?ⁿ. We prove interpolation theorems for the subadditive operators of weak type (?₀, ?₀) bounded in L _∞(?ⁿ) and subadditive operators of weak types (?₀, ?₀) and (?₁, ?₁) in L _?(? ⁿ ) under some assumptions on the nonnegative and increasing functions ?(x) on [0, ∞). We also obtain some interpolation theorems for the linear operators of weak type (?₀, ?₀) bounded from L _∞(?ⁿ) to BMO(? ⁿ). For the restrictions of these operators to the set of characteristic functions of Lebesgue measurable sets, we establish some estimates for rearrangements of moduli of their values; deriving a consequence, we obtain a theorem on the boundedness of operators in rearrangement-invariant spaces.     

The Incidence Divisor of Cycles in Projective Space

For nonnegative integers a, b with a + b + 1 = n, we show the incidence locus has the structure of an effective Cartier divisor in the product of Chow varieties 𝒞 _a (? ⁿ ) × 𝒞 _b (? ⁿ ).     

On the mutually non isomorphic ℓp(ℓq) spaces

We extend a result of Pe?czyński showing that {?_p(?_q): 1 ≤ p, q ≤ ∞} is a family of mutually non isomorphic Banach spaces. Some results on complemented subspaces of ?_p(?_q) are also given.     

The effect of a penalty term involving higher order derivatives on the distribution of phases in an elastic medium with a two‐well elastic potential

We consider the problem of minimizing 0<p<1, h∈?, σ>0, among functions u:?^d?Ω→?^d, u_∣?Ω=0, and measurable characteristic functions χ:Ω→?. Here ?⁺_h, ?^?, denote quadratic potentials defined on the space of all symmetric d×d matrices, h is the minimum energy of ?⁺_h and ε(u) denotes the symmetric gradient of the displacement field. An equilibrium state û, χ?, of I [·,·,h, σ] is termed one‐phase if χ?≡0 or χ?≡1, two‐phase otherwise. We investigate the way in which the distribution of phases is affected by the choice of the parameters h and σ. Copyright 2002 John Wiley & Sons, Ltd.     

Estimates of the smoothness of dyadic orthogonal wavelets of Daubechies type

Suppose that ω(φ, ·) is the dyadic modulus of continuity of a compactly supported function φ in L ²(?⁺) satisfying a scaling equation with 2 ⁿ coefficients. Denote by α _φ the supremum for values of α > 0 such that the inequality ω(φ, 2^?j ) ≤ C2 ^?αj holds for all j ∈ ?. For the cases n = 3 and n = 4, we study the scaling functions φ generating multiresolution analyses in L ²(?₊) and the exact values of α _φ are calculated for these functions. It is noted that the smoothness of the dyadic orthogonal wavelet in L ²(?₊) corresponding to the scaling function φ coincides with α _φ .     

Timelike surfaces of constant mean curvature ±1 in anti-de Sitter 3-space ℍ3 1(−1)

It is shown that timelike surfaces of constant mean curvature ± in anti-de Sitter 3-space ?³ ₁(?1) can be constructed from a pair of Lorentz holomorphic and Lorentz antiholomorphic null curves in ?SL₂? via Bryant type representation formulae. These Bryant type representation formulae are used to investigate an explicit one-to-one correspondence, the so-called Lawson–Guichard correspondence, between timelike surfaces of constant mean curvature ± 1 and timelike minimal surfaces in Minkowski 3-space E ³ ₁. The hyperbolic Gauß map of timelike surfaces in ?³ ₁(?1), which is a close analogue of the classical Gauß map is considered. It is discussed that the hyperbolic Gauß map plays an important role in the study of timelike surfaces of constant mean curvature ± 1 in ?³ ₁(?1). In particular, the relationship between the Lorentz holomorphicity of the hyperbolic Gauß map and timelike surface of constant mean curvature ± 1 in ?³ ₁(?1) is studied.