New considerations on Hilbert's problem (Q2762825)

A new point of view on the old Hilbert problem for the unit circle in the complex plane is pointed out. The boundary condition \(a(s)u(s)+b(s)v(s)=c(s)\) ; \(s \in [-\pi,\pi]\), where \(F(z)=u(x,y)+iv(x,y)\) is the solution of the boundary value problem, and \(a\), \(b\), and \(c\) are Hölder continuous functions, is considered with respect to the following two cases: (a) \(a(s)\) and \(b(s)\) are inside conjugated, (b) \(a(s)\) and \(b(s)\) are arbitrary. The functions \(a(s)\) and \(b(s)\) are inside conjugated if there exist a function \(f(z)\), holmorphic in the unit disk \(S\), and continuous on its closure \(\overline S\), such that \(f(e^{is})=a(s)+i b(s)\). The method works by applying to a previous result of the author which characterizes of inside conjugated functions. In this way, the difficulties encountered in the classical approach, due to singular integral equations, are overcome.