A stationary heat conduction problem in low dimensional sets in $${\mathbb {R}}^N$$RN

We study a basic linear elliptic equation on a lower dimensional rectifiable set S in $${\mathbb {R}}^N$$ with the Neumann boundary data. Set S is a support of a finite Borel measure $$\mu $$. We will use the measure theoretic tools to interpret the equation and the Neumann boundary condition. For this purpose we recall the Sobolev-type space dependent on the measure $$\mu $$. We establish existence and uniqueness of weak solutions provided that an appropriate source term is given.