Riemann-Hilbert problem

A closely related problem is given a linear relationship between the real and imaginary parts of the function on the boundaries of a domain in which it is analytic, find the function throughout the domain. This is termed the Riemann-Hilbert problem by Galin (1961) and also here. 

So, in the case of the Riemann-Hilbert problem, a function is analytic in a given domain bounded by a particular contour, and information is supplied about the limiting values of the function on this contour. In the case of the Hilbert problem, the region of analyticity contains open or closed contours on which the function is not defined. These are cuts in the region of analyticity across which the function is discontinuous. The supplied information is a relationship between the limiting values on either side of these cuts or internal boundaries. 

Plane elastic, and in a partial sense, viscoelastic boundary value problems can often be phrased in terms of either a Hilbert or a Riemann-Hilbert problem. Galin (1961) for example, bases his whole approach on the latter method. On the other hand, Muskhelishvili (1963) manages to cast various boundary value problems in the form of Hilbert problems, thereby solving them. Gakhov (1966) discusses the connection between the two problems. The Hilbert approach is adopted here, on the grounds that the theory is a little easier, though this may be a matter of taste. 

The Riemann-Hilbert problem is therefore not discussed. The treatment of the Hilbert problem is not very general, being confined in particular to the case of constant coefficients. The reader is referred to the standard texts mentioned above for more complete treatments. 

Our object is to apply (2.8.9) to problems involving loads on viscoelastic halfspaces. For such problems, it is desirable to re-express these equations in an alternative form [Muskhelishvili (1963)] which facilitates reduction to a Hilbert problem. Let the material occupy the upper half-plane > >0 so that (piz, t) is analytic in this region. It is convenient to extend the region of analyticity of (p(z, t) to the lower half-plane also. Then, as we shall see, it is possible to explore the discontinuities in this function across the real axis, which gives a Hilbert problem. Another approach, possibly more direct, is that of Galin, mentioned previously, which leads to problems of the Riemann-Hilbert type. These however are somewhat more difficult to deal with, from a mathematical point of view. 

The problem of the linear relationship may be stated as follows given the equation connecting the boundary values of an analytic function on a line of discontinuity together with some condition on the function at infinity, determine the analytic function, either uniquely or very nearly uniquely. This has been termed the Hilbert problem by Muskhelishvili (1953, 1963) and the Riemann problem by Gakhov (1966). We adopt the former terminology here. 

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