Analysis of conditions of solvability for a class of space riemann problems

The torus related Riemann problem

“…But φ −− (z 1 , ∞) and φ −− (∞, z 2 ) are not necessarily zero as it was needed in [11], except at most φ −− (∞, ∞) = 0.…”

“…More deeper inside knowledge is needed. Because of these difficulties, although there were some papers about the Shilov boundary related special inhomogeneous Riemann problem in C n (n > 1) [5,11,15], no one has given a solution which is constructed by the canonical function for true higher dimensional torus domains so far, except [10,12] for bi-disc domains. However, the latter results have been found incorrect or inadequate [2].…”

“…For this reason applying Fourier series method and analyzing the structure of boundary values of analytic functions, some complementary concepts about the Noether condition [11], for higher dimensional torus domains have been clarified [1,20], which forced previous discussions [2,12] to make some compromises on the properties of analytic functions in general and so lead to some artificial assumptions. The rearranged form of the boundary values of holomorphic functions in polydomains by the modified Cauchy kernel [1,20] and the well-posed formulation of the Riemann problem are the key factors for the establishment of the connection between the two problems for polydomains.…”

“…To apply these techniques for torus domains the problem was considered variable by variable so that one inner and its outer domain are always available. Thus because of the Noether condition for analytic functions of the outer domain in the plane, analytic functions of some torus domains get more strict restrictions [11] than necessary and adequate [1,20].…”

The Riemann–Hilbert problem for certain poly domains and its connection to the Riemann problem

“…About the Riemann problem for poly disc and poly domains there are some results in literature [3,6,[14][15][16]19,26,28]. About the Riemann-Hilbert problem for poly domains there is nothing known, except for the poly disc D n [2,4].…”