Traces, extensions, co-normal derivatives and solution regularity of elliptic systems with smooth and non-smooth coefficients

For functions from the Sobolev space $H^s(\mathrm{\Omega })$, 1/2<s<3/2, definitions of non-unique generalised and unique canonical co-normal derivative are considered, which are related to possible extensions of a partial differential operator and its right hand side from the domain $\mathrm{\Omega }$, where they are prescribed, to the domain boundary, where they are not. Revision of the boundary value problem settings, which makes them insensitive to the co-normal derivative inherent non-uniqueness are given. Some new facts about trace operator estimates, Sobolev spaces characterisations, and solution regularity of PDEs with non-smooth coefficients are also presented.