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Elliptic systems of pseudodifferential equations in the refined scale on a closed manifold

Mikhailets V. A., Murach A. A.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

Vol. 56, no 3-4

213-224

We study a system of pseudodifferential equations which is elliptic in the Petrovskiî sense on a closed smooth manifold. We prove that the operator generated by the system is a Fredholm operator in a refined two-sided scale of Hilbert function spaces. Elements of this scale are special isotropic spaces of Hörmander-Volevich-Paneah.

The reactance wave diffraction problem by a strip in a scale of Bessel potential spaces

Castro L. P., Natroshvili D.

Opuscula Mathematica

2006

Vol. 26, no. 2

289-303

We consider a boundary-transmission problem for the Helmholtz equation, in a Bessel potential space setting, which arises within the context of wave diffraction theory. The boundary under consideration consists of a strip, and certain reactance conditions are assumed on it. Operator theoretical methods are used to deal with the problem and, as a consequence, several convolution type operators are constructed and associated to the problem. At the end, the well-posedness of the problem is shown for a range of regularity orders of the Bessel potential spaces, and for a set of possible reactance numbers (dependent on the wave number).