Solvability of a nonlocal boundary value problem and the smoothness of the domain (Q2388049)

In the case of a smooth domain the nonlocal boundary value problem is posed as follows. Let \(\Omega\) be a simply connected bounded \(C^2\) smooth domain in \(\mathbb{R}^n\), \(n\geq 2\), \(\partial\Omega\) its boundary, and let \(S\) be a closed smooth surface in \(\Omega\) \((S\subset\Omega)\). Next, let \(z=z(x)\) be a \(C^2\)-diffeomorphism of \(\partial\Omega\) onto \(S\). For an arbitrary function \(f\in C(\partial\Omega)\), find a harmonic function \(V\), \(V\in C(\Omega)\), such that \[ V(x)-V\bigl(z(x)\bigr)= f(x),\quad x\in \partial\Omega. \] The author gives a necessary and sufficient condition for solving this problem.