By the low-dimensional structure we understand a closed subset of Euclidean space possessing some geometrical nature. The class includes, for instance, CW-complexes, families of manifolds, stratified manifolds etc. We very briefly recall the current state of the art of PDEs on such structures.

We present the most characteristic notions of the variational calculus and elliptic PDEs in this setting. The main part of the talk is focused on a discussion of our elliptic regularity results. We sketch the proof of upgrading the regularity of weak solutions to ensure membership in the domain of the low-dimensional counterpart of the second-order derivative. Besides that, we plan to mention a few words concerning the low-dimensional parabolic problems.