A generalized Riemann-Hilbert problem and its applications (Q5929645)

The authors develop an existene and uniqueness theory for the Riemann-Hilbert problem of finding a function \(\varphi(z)\) analytic in the open set \(D\) and continuous in the closed domain \(\overline{D}=D\cup\Gamma_1\cup\Gamma_2\), \(\Gamma_1\) being the circle \(|z|=r_0\) and \(\Gamma_2\) a smooth simple closed curve such that \(\min\{|z|:z\in\Gamma_2\}>r_0\), which satisfies \[ \text{ Re} [a(z)\varphi(z)]=f(z),\qquad z\in\Gamma_2, \] \[ \varphi(z)-\varphi(\overline{z})=g(z),\qquad z\in\Gamma_1,\;\text{ Im} z\geq 0. \] Here \(a(z)\neq 0\) for every \(z\in\Gamma_1\), \(g(r_0)=g(-r_0)=0\), \(f(z)\) is real-valued, and \(a,f,z\) are Hölder continuous. No clear-cut theorems are provided, but instead the solution of the above problem is reduced to the solution of such a problem for \(a(z)\equiv 1\) and \(g(z)\equiv 0\) and the latter problem is solved. The results are specialized to the case where \(\Gamma_2=\{z:|z|=r_1\}\) for some \(r_1>r_0\). As an application, the Laplace equation on bounded regions in the complex plane whose boundary is an analytic curve, assuming (inhomogeneous) Neumann-type boundary conditions, is reduced to the above Riemann-Hilbert problem. In some special cases a complete solution is obtained.