Volume 14, Issue 1

We study the analogue of the Cauchy transform for the theory of solutions of the Hodge/de Rham system in the case of a rectifiable surface of integration which additionally satisfies an Ahlfors/David regularity condition and we prove the Cauchy integral formula, the Plemelj/Privalov theorem and the Sokhotski/Plemelj theorem for it, as well as the necessary and sufficient condition for the possibility to extend a given 𝑘-form from such a surface to a harmonic 𝑘-form in the domain. A formula for the square of the singular Cauchy transform is given. The proofs of all these facts are based on a close relation between algebra-valued null-solutions of the Dirac operator in the Euclidean space and hyperholomorphic functions of Clifford analysis.

We improve a recent result of Y. Gotoh [Pacific J. Math 201: 289-307, 2001] who has established a precise relation between the constants in P. W. Jones theorem [Ark. Mat. 21: 229-231, 1983] about homeomorphisms of the line preserving BMO . We also give an explicit bound for a distance to L ∞ after composition.

On the basis of a higher order integral representation formula related to the polyharmonic differential operator and obtained through a certain polyharmonic Green function, a Dirichlet problem is explicitly solved in the upper half plane.

The monomiality principle was introduced (see [Dattoli, Hermite–Bessel and Laguerre–Bessel functions: a by-product of the monomiality principle: 147–164, Aracne, 2000] and the references therein) in order to derive the properties of special or generalized polynomials starting from the corresponding ones of monomials. We show a general technique of extending the monomiality approach to multi-index polynomials in several variables. Application of this technique to the case of Hermite, Laguerre-type and mixed-type (i.e., between Laguerre and Hermite) polynomials is given.

We consider a multidimensional analogue of the Darboux problem for wave equations with power nonlinearity. Depending on the spatial dimension of an equation, a power nonlinearity exponent and the sign in front of a nonlinear term, it is proved that the Darboux problem is globally solvable in some cases, but has no global solution in other cases though the local solvability of this problem remains in force.

Let { ω 𝑘 } be a complete system of polynomial solutions of the elliptic equation ∑ | α |⩽2𝑚 a α 𝐷 α 𝑢 = 0, a α being real constants. We give necessary and sufficient conditions for the completeness of the system in [𝐿 𝑝 (∂Ω)] 𝑚 , where Ω ⊂ is a bounded domain such that is connected and ∂Ω ∈ 𝐶 1 .

Motivated by recent studies of the properties of new classes of polynomials constructed in terms of quasi-monomials, certain generating relations involving Hermite–Tricomi functions are obtained. To accomplish this we use the representation 𝑄(𝑤, 𝑚 0 ) of the 3-dimensional Lie algebra 𝑇 3 . Some special cases are also discussed.

In this paper, the near-field approximation of the acoustic Green's function in a two-layer waveguide is constructed by using a variation of the method of Ahluwalia and Keller [Exact and asymptotic representations of the sound field in a stratified ocean, Springer, 1977]. The relation between the constructed multiple-scattering representation (suitable for near-field) and the Hankel transform representation (suitable for mid-range) is also discussed in this paper. The construction scheme presented in this paper can be generalized for an N-layer waveguide.

A constructive approach is presented to treat an initial boundary value problem for isothermal Navier–Stokes equations. It is based on a characteristics (Lagrangean) approximation locally in time and a boundary integral equation method via nonstationary potentials. As a basic problem, the latter leads to a Volterra integral equation of first kind which is proved to be uniquely solvable and even coercive in some anisotropic Sobolev spaces. The solution depends continuously upon the data and can be constructed by a quasioptimal Galerkin procedure.

For an infinite-dimensional separable Hilbert space 𝐻, the problem of measurability of additive functionals 𝑓 : 𝐻 → 𝐑 with respect to various extensions of σ -finite diffused Borel measures on 𝐻 is discussed. It is shown that there exists an everywhere discontinuous additive functional 𝑓 on 𝐻 such that, for any σ -finite diffused Borel measure μ on 𝐻, this 𝑓 can be made measurable with respect to an appropriate extension of μ . Special consideration is given to the case where μ is invariant or quasiinvariant under a subgroup of 𝐻.

We consider BVPs for strongly elliptic systems of order 2𝑙 with the boundary conditions of order 𝑙 + 𝑛, 𝑛 ⩾ 0. By representing the solution by means of a simple layer potential of order 𝑛 and by imposing the boundary conditions, we get a singular integral system which is of regular type if and only if the boundary operator satisfies the Lopatinskiĭ condition and which can be solved if suitable compatibility conditions are satisfied. An explicit formula for computing the index of the BVP is given.

We construct a sequence of quadratic weighted energy forms in an open domain of the plane, that 𝑀-converges to an energy form with a singular fractal term. The weights belong to the Muckenoupt class 𝐴 2 and have pointwise singularities. The result implies the spectral convergence of a sequence of second-order weighted elliptic operators in divergence form in the plane to a singular elliptic operator with a second order fractal term.

In case a non-linear differential equation is reduced to a fixed-point problem, one has to apply a fixed-point theorem to a suitably chosen subset of the underlying function space. Generally speaking, in view of the non-linearity of the differential equation the restrictions for the data of the problem will be the more extensive the greater the chosen subset of the function space. The problem is to find an optimal subset leading to a domain of existence for the desired solution being as large as possible. The present paper will discuss this question for a Goursat problem.