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

Abstract

For functions from a Sobolev space, 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, 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.