Stacjonarne zagadnienie przepływu ciepła na niżej wymiarowych zbiorach prostowalnych w $R^N$

I will discuss an elementary linear elliptic equation on a lower dimensional rectifiable structure in $R^N$ with Neumann boundary data. The set may be described by means of a finite Borel measure µ supported on it. This allows us to reformulate the equation and the boundary condition and to establish existence and uniqueness of a weak solution via a variational method. The setting requires an appropriate definition of a Sobolev-type space dependent on the measure µ and an appropriate Poincaré-type inequality. I will present examples of structures that are not manifolds and which do not support a global Poincaré inequality, yet which are admissible for our setting.