Solution regularity and co-normal derivatives for elliptic systems with non-smooth coefficients on Lipschitz domains

Abstract

Elliptic PDE systems of the second order with coefficients from L∞ or Holder-Lipschitz spaces are considered in the paper. Continuity of the operators in corresponding Sobolev spaces is stated and the internal (local) solution regularity theorems are generalized to the non-smooth coefficient case. For functions from the Sobolev space H^s(Omega), 0.5<s<1.5, definitions of non-unique generalized and unique canonical co-normal derivative are considered, which are related to possible extensions of a partial differential operator and the PDE right hand side from the domain $\Omega$ to its boundary. It is proved that the canonical co-normal derivatives coincide with the classical ones when both exist. A generalization of the boundary value problem settings, which makes them insensitive to the co-normal derivative inherent non-uniqueness is given.