  This paper aims to establish counterparts of fundamental regularity
statements for solutions to elliptic equations in the setting of
low-dimensional structures such as, for instance, glued manifolds or
CW-complexes. The main result proves additional Sobolev-type regularity of weak
solutions to elliptic problems defined on the low-dimensional structure. Due to
the non-standard geometry of a domain, we propose a new approach based on
combining suitable extensions of functions supported on the thin structure with
uniform bounds for difference quotients. We also derive several important
conclusions from that result, namely global continuity of weak solutions and we
address the correspondence between the low-dimensional weak problems and the
second-order operators widely applied in the framework developed for
applications in variational problems.
