Serrin's overdetermined problem and sharp harmonic quadrature identities in the plane

We study a weak formulation of Serrin's overdetermined boundary value problem in planar Jordan domains with rectifiable boundary. Our first result establishes that, within the class of rectifiable Jordan Smirnov domains, the corresponding harmonic quadrature identity, equivalent to Serrin's overdetermined problem, necessarily implies that the domain is a disk. Subsequently, we construct a family of rectifiable, non-Smirnov Jordan domains that nonetheless satisfy the same quadrature identity, thereby demonstrating the sharpness of the Smirnov regularity assumption. Consequently, there exists a nontrivial Jordan domain with rectifiable boundary satisfying the weak formulation of Serrin's overdetermined system in $\mathbb R^2$.