On solvability of the jump problem (Q1025037)

Let \(\Gamma\) be a closed Jordan curve in the plane \(\mathbb{C}\). It divides \(\overline \mathbb{C}\) into a finite domain \(D^+\) and domain \(D^-\) containing \(\infty\). Let \(f(t)\) be defined on \(\Gamma\). We look for a function \(\varphi(z)\) holomorphic on \(\overline\mathbb{C}\setminus\Gamma\), \(\varphi(\infty)=0\), such that the boundary values \[ \varphi^+(t):=\lim_{D^+\ni z\to t}\varphi(z),\;\quad\varphi^-(t):=\lim_{D^-\ni z\to t}\varphi(z) \] exist for any \(t\in\Gamma\) and such that \[ \varphi^+(t)-\varphi^-(t)=f(t) \quad\text{for }t\in\Gamma. \] This is the so-called jump problem. It is known that for a piecewise smooth curve \(\Gamma\), and a jump function \(f(t)\) satisfying a Hölder condition, the jump problem has a unique solution by the Cauchy integral \[ \varphi(z)=\frac{1}{2\pi i} \int_\Gamma\frac{f(\zeta)}{\zeta-z}d \zeta. \] In this paper, the author looks for a solution of the jump problem for curves \(\Gamma\) which are not rectifiable. To solve this problem, special polygonal approximations \(\Gamma_n\) of the curve \(\Gamma\) and special approximations \(f_n(t)\) of the jump function are constructed. Thus the solution has the form \[ \varphi(z)=\lim_{n\to\infty}\frac{1}{2\pi i} \int_{\Gamma_n}\frac{f_n(\zeta)}{\zeta-z}d\zeta. \]