Roland Duduchava

The Green formula and layer potentials

The explicit form of all possible variants of the Green formula is described for a boundary value problem when the “basic” operator is an arbitrary partial differential operator with variable matrix coefficients and the “boundary” operators are quasi-normal with vector-coefficients. If the system possesses a fundamental solution, a representation formula for the solution is derived and boundedness properties of the relevant layer potentials, mapping function spaces on the boundary (Bessel potential, Besov, Zygmund spaces) into appropriate weighted function spaces on the domain are established. We conclude by discussing some closely related topics: traces of functions from weighted spaces, traces of potential-type functions, Plemelji formulae, Calderón projections, and minimal smoothness requirements for the surface and coefficients.

On algebras generated by convolutions and discontinuous functions

Banach algebras generated by Fourier and Mellin convolution operators with discontinuous presymbols and by discontinuous functions in Lp (IR+, xα) spaces with weight are investigated. The Fredholm properties are characterized by a symbol calculus and an index formula for such operators is presented. These results were obtained by H. O. Cordes in [3] for the case p=2, α=0 and presymbols, which are discontinuous only at infinity and generalized in [20] for 1<p<∞.Up to now, the most extensive study of the operators under consideration was carried out in [4–8] (the case α=0, 1<p<∞), but even these results are revised here.

Singular Integral Equations on Piecewise Smooth Curves in Spaces of Smooth Functions

We prove the boundedness of the Cauchy singular integral operator in modified weighted Sobolev \( \mathbb{K}\mathbb{W}_{p}^{m}(\Gamma ,\rho ) \), Holder-Zygmund \( \mathbb{K}\mathbb{Z}_{\mu }^{0}(\Gamma ,\rho ) \) Bessel potential \( \mathbb{K}\mathbb{H}_{p}^{s}(\Gamma ,\rho ) \) and Besov \( \mathbb{K}\mathbb{B}_{{p,q}}^{s}(\Gamma ,\rho ) \) spaces under the assumption that the smoothness parameters m,μ,s are large. The underlying contour Γ is piecewise smooth with angular points and even with cusps. We obtain Fredholm criteria and an index formula for singular integral equations with piecewise smooth coefficients and complex conjugation in these spaces provided the underlying contour has angular points but no cusps. The Fredholm property and the index turn out to be independent of the integer parts of the smoothness parameters m,µ,s. The results are applied to an oblique derivative problem (the Poincare problem) in plane domains with angular points and peaks on the boundary.

Bessel potential operators for the quarter-plane

Bessel potential operators of order r are defined as certain translation invariant operators, which act bijectively between Sobolev spaces . This paper presents the Fourier symbol of a two-dimensional Bessel po-tential operator, which has the invariance property: its preserves the support of functions within the first quadrant. As an application, the solution of a well-known diffraction problem is given in explicit form.

Mixed boundary value problems for the Helmholtz equation in arbitrary 2D-sectors

Abstract. The purpose of the present research is to investigate the mixed Dirichlet–Neumann boundary value problems for the Helmholtz equation in a 2D domain with finite number of non-cuspidal angular points on the boundary. Using localization, the problem is reduced in [Proc. A. Razmadze Math. Inst. 162 (2013), 37–44] to model problems in plane angles of magnitude , . In the present paper, we apply the potential method and reduce the model mixed BVP (with Dirichlet–Neumann conditions on the boundary) to an equivalent boundary integral equation (BIE) of Mellin convolution type. Applying the recent results on Mellin convolution equations with meromorphic kernels in Bessel potential and Sobolev–Slobodeckij (Besov) spaces obtained by V. Didenko and R. Duduchava and by R. Duduchava, the unique solvability criteria (Fredholm criteria) of the above mentioned mixed BVP are obtained in classical finite energy space and also in non-classical Bessel potential spaces when . The problem has been tackled before only for angular domains of magnitude for rational angles .

LOCALIZATION AND MINIMAL NORMALIZATION OF SOME BASIC MIXED BOUNDARY VALUE PROBLEMS

We consider a class of mixed boundary value problems in spaces of Bessel potentials. By localization, an operator L associated with the BVP is related through operator matrix identities to a family of pseudodifferential operators which leads to a Fredholm criterion for L. But particular attention is devoted to the non-Fredholm case where the image of L is not closed. Minimal normalization, which means a certain minimal change of the spaces under consideration such that either the continuous extension of L or the image restriction, respectively, is normally solvable, leads to modified spaces of Bessel potentials. These can be characterized in a physically relevant sense and seen to be closely related to operators with transmission property (domain normalization) or to problems with compatibility conditions for the data (image normalization), respectively.

Asymmetric Factorizations of Matrix Functions on the Real Line

We indicate a criterion for some classes of continuous matrix functions on the real line with a jump at infinity to admit both, a classical right and an asymmetric factorization. It yields the existence of generalized inverses of matrix Wiener-Hopf plus Hankel operators and provides precise information about the asymptotic behavior of the factors at infinity and of the solutions to the corresponding equations at the origin.

Wiener-Hope equations with the transmission property

AbstractConditions on the kernel of the classical Wiener-Hopf equation are obtained which provide the same smoothness of any solution in terms of its inclusion in different spaces of smooth functions such as

Mellin convolution operators in Bessel potential spaces

C^s \left( {\mathbb{R}^ + } \right), Z^s \left( {\mathbb{R}^ + } \right), H_p^s \left( {\mathbb{R}^ + } \right), and B_{p, q}^s \left( {\mathbb{R}^ + } \right)