Given two harmonic functions $u_{+}(x,y)$, $u_{-}(x,y)$ defined on opposite sides
of the $y$-axis in $\mathbb{R}^2$ and
periodic in $y$, we consider the problem of constructing a
{\it family of gluing elliptic functions}, i.e. a family of functions $u_{\epsilon}(x,y)$
of class ${\mathcal C}^{1,1}$ that coincide with
$u_+$ and $u_-$ outside neighborhoods of the $y$-axis of width less than $\epsilon$ and
are solutions to linear, uniformly elliptic equations without zero order terms. We first
show that not always there is such a family and we
give a necessary condition for its existence. Then we
give a sufficient condition for
the existence of a family of gluing elliptic functions and a way for its construction.