[en] In the present paper, with the aid of the techniques of micro-local analysis, a regularity theorem of the solutions to Dirichlet problem for a class of nonlinear degenerate elliptic equations is given. (author). 8 refs

[en] We consider an inverse problem in elastodynamics arising in seismic imaging. We prove locally uniqueness of the density of a non-homogeneous, isotropic elastic body from measurements taken on a part of the boundary. We measure the Dirichlet to Neumann map, only on a part of the boundary, corresponding to the isotropic elasticity equation of a 3D object. In earlier works it has been shown that one can determine the sheer and compressional speeds on a neighborhood of the part of the boundary (accessible part) where the measurements have been taken. In this article we show that one can determine the density of the medium as well, on a neighborhood of the accessible part of the boundary. (paper)

[en] New necessary and sufficient conditions for the uniform approximability of functions by polyanalytic polynomials and polyanalytic rational functions on compact subsets of the plane are established. Connections between these approximation problems and the Dirichlet problem for bianalytic functions are also analysed

[en] The dynamics of the behaviour of the absolute values of the Dirichlet kernels is described: the discrete dynamics for the Dirichlet kernel of the Walsh system and the continuous dynamics for the generalized Walsh-Dirichlet kernel. Estimates of the p-norms of the Dirichlet kernels are obtained. The concept of generalized Lebesgue constant is introduced and the corresponding formulae are found, which generalize Fine's formulae for the Lebesgue constants. These results hold not only for the Walsh system in the Paley enumeration, but also for rearrangements of the Walsh systems, including linear and piecewise linear ones.

[en] Upper bounds are obtained for solutions of the Dirichlet problem for pseudodifferential elliptic equations where the right-hand side has compact support. In domains with non-compact boundary they characterise the behaviour of solutions at infinity in its dependence on the geometric properties of the domain. For unbounded domains where the boundary has irregular behaviour, it is shown that these bounds may be more efficient than the bounds that are already known for second-order elliptic equations. For second-order elliptic equations in a broad class of domains of revolution these bounds are shown to be sharp. Bibliography: 17 titles.

[en] We consider the stability issue of the inverse conductivity problem for a conformal class of anisotropic conductivities in terms of the local Dirichlet–Neumann map. We extend here the stability result obtained by Alessandrini and Vessella (Alessandrini G and Vessella S 2005 Lipschitz stability for the inverse conductivity problem Adv. Appl. Math. 35 207–241), where the authors considered the piecewise constant isotropic case. (paper)

[en] This paper deals with analytic properties of certain Dirichlet series considered by Davenport and Heilbronn and by Titchmarsh. Bibliography: 28 titles.

[en] We study the behavior of solutions to the Dirichlet problem for the p(x)-Laplacian with a continuous boundary function. We prove the existence of a weak solution under the assumption that p is separated from 1 and ∞. We present a necessary and sufficient Wiener type condition for regularity of a boundary point provided that the exponent p has the logarithmic modulus of continuity at this point.

[en] The aim of this paper is using an elementary method and the properties of the Bernoulli polynomials to establish a close relationship between the Euler numbers of the second kind $E_n^{\ast <!--\; ???\; -->}$ and the Dirichlet L-function $L(s,\mathrm{\chi <!--\; ??\; -->})$. At the same time, we also prove a new congruence for the Euler numbers $E_n$. That is, for any prime $p\equiv <!--\; ???\; -->1mod8$, we have $E_{\frac{p-<!--\; ???\; -->3}{2}}\equiv <!--\; ???\; -->0modp$. As an application of our result, we give a new recursive formula for one kind of Dirichlet L-functions.

[en] We show that the knowledge of the Dirichlet-to-Neumann map on the boundary of a bounded open set in ${\mathbb{R}}^n$, $n⩾<!--\; ???\; -->3$, for the perturbed polyharmonic operator ${(-<!--\; ???\; -->\mathrm{\Delta <!--\; ??\; -->})}^m+A\cdot <!--\; ???\; -->D+q$, $m⩾<!--\; ???\; -->2$, with $2n><!--\; >\; -->m$, $A\in <!--\; ???\; -->W^{-<!--\; ???\; -->\frac{m-<!--\; ???\; -->2}{2},\frac{2n}{m}}$ and $q\in <!--\; ???\; -->W^{-<!--\; ???\; -->\frac{m}{2},\frac{2n}{m}}$, determines the potentials A and q in the set uniquely. The proof is based on a Carleman estimate with linear weights and with a gain of two derivatives and on the property of products of functions in Sobolev spaces. (paper)