Carleman boundary value problem

A boundary value problem for analytic functions involving a shift which reverses the direction of traversing the boundary. It was first considered by T. Carleman [1]. Let $ L $ be a simple closed Lyapunov curve (cf. Lyapunov surfaces and curves) in the complex $ z $- plane and let $ D $ be the finite domain bounded by $ L $. Let $ \alpha (t) $ be a given complex-valued function on $ L $ inducing a one-to-one mapping of $ L $ onto itself reversing the direction of traversing $ L $ and satisfying additionally the Carleman condition:

$$ \tag{* } \alpha [ \alpha (t)] = t,\ \ t \in L $$

(it is further supposed that the derivative $ \alpha ^ \prime (t) $ satisfies a Hölder condition). Then the Carleman boundary value problem consists in finding a function $ \Phi (z) $, analytic in $ D $ except for a finite number of poles, continuous on $ D \cup L $ and subject to the boundary condition

$$ \Phi [ \alpha (t)] = \ G (t) \Phi (t) + g (t),\ \ t \in L, $$

where the functions $ G (t) $ and $ g (t) $ given on $ L $ satisfy a Hölder condition and $ G (t) \neq 0 $ on $ L $.

The Carleman boundary value problem has also been studied under the condition

$$ \alpha ^ {m} (t) = t,\ \alpha ^ {1} (t) = \alpha (t),\ \ \alpha ^ {k} (t) = \ \alpha ( \alpha ^ {k - 1 } (t)),\ \ k = 2 \dots m, $$

which is more general than (*), and so has the Carleman boundary value problem for several unknown functions (see [2], [3]).

References

[1]	T. Carleman, "Sur la théorie des équations intégrales et ses applications" , Verh. Internat. Mathematiker Kongress. Zürich, 1932 , 1 , O. Füssli (1932) pp. 138–151
[2]	N.I. Muskhelishvili, "Singular integral equations" , Wolters-Noordhoff (1977) (Translated from Russian)
[3]	N.P. Vekua, "Systems of singular integral equations and some boundary value problems" , Moscow (1970) (In Russian)