  We prove a general result characterizing a specific class of Serrin domains
as supports of unbounded and periodic constant mean curvature graphs. We apply
this result to prove the existence of a family of unbounded periodic constant
mean curvature graphs, each supported by a Serrin domain and intersecting its
boundary orthogonally, up to a translation. We also show that the underlying
Serrin domains are calibrable and Cheeger in a suitable sens, and they solve
the 1-Laplacian equation.
