  The classical Serrin's overdetermined theorem states that a $C^2$ bounded
domain, which admits a function with constant Laplacian that satisfies both
constant Dirichlet and Neumann boundary conditions, must necessarily be a ball.
While extensions of this theorem to non-smooth domains have been explored since
the 1990s, the applicability of Serrin's theorem to Lipschitz domains remained
unresolved. This paper answers this open question affirmatively. Actually, our
approach shows that the result holds for domains that are sets of finite
perimeter with a uniform upper bound on the density, and it also allows for
slit discontinuities.
