Open AccessThe Riemann – Hilbert boundary value problem for elliptic systems of the orthogonal type in R<sup>3</sup>
Author(s) -
A. I. Basik,
E. V. Hrytsuk,
T. A. Hrytsuk
Publication year - 2020
Publication title -
vescì nacyânalʹnaj akadèmìì navuk belarusì. seryâ fìzìka-matèmatyčnyh navuk
Language(s) - English
Resource type - Journals
eISSN - 2524-2415
pISSN - 1561-2430
DOI - 10.29235/1561-2430-2020-56-1-7-16
Subject(s) - mathematics , boundary value problem , mathematical analysis , riemann–hilbert problem , elliptic boundary value problem , boundary (topology) , pure mathematics , type (biology) , elliptic operator , operator (biology) , robin boundary condition , mixed boundary condition , ecology , biochemistry , chemistry , repressor , gene , transcription factor , biology
In this paper, a class of elliptic systems of four 1st order differential equations of the orthogonal type in R 3 is considered. For such systems we study the issue of regularizability of the Riemann – Hilbert boundary value problem in an arbitrary limited simply-connected region with a smooth boundary in R 3 . Using the coefficients of the elliptic system and the matrix of the boundary operator, a special vector field is constructed, and its not entering the tangent plane in any point of the boundary provides the Lopatinski condition of the regularizability of the boundary value problem. The obtained condition permits to prove that the set of regularizable Riemann – Hilbert boundary value problems for the considered class of systems has two components of homotopic connectedness, and the index of an arbitrary regularizable problem equals to minus one.