On the Dirichlet and the Riemann-Hilbert problem on Möbius strips (Q686858)

The goal of the present paper is to study the Dirichlet problem and the Riemann-Hilbert problem on Möbius strips \({\mathcal M}\). To derive a Fredholm theory the author has to deal with the following problems: 1. Functions on \({\mathcal M}\): these are the functions \(g\) on the annulus \({\mathcal R}=\{z\in\mathbb{C}\): \({1\over R}<| z|<R\}\); 2. Harmonic functions on \({\mathcal M}\): the real part \(\text{Re }f\) of holomorphic functions \(f\) on \({\mathcal R}\) can be identified with harmonic functions on \({\mathcal M}\); 3. Function spaces on \({\mathcal M}\): function spaces of harmonic functions on \({\mathcal M}\) and corresponding trace spaces on \(\partial {\mathcal M}\) of Sobolev type are introduced; 4. The study of special operators in scales of Sobolev spaces on \({\mathcal M}\). After these preparations the author studies the Laplacian and the Riemann-Hilbert operator on Sobolev spaces of harmonic functions on \({\mathcal M}\). The properties of these operators lead to statements (existence, uniqueness) for the Dirichlet problem and (Fredholm property, determination of index) for the Riemann-Hilbert problem.