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

*(English)*Zbl 1308.35086Summary: Elliptic PDE systems of the second order with coefficients from \(L_{\infty }\) or HĂ¶lder-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 \(H^{s}(\Omega ),\frac {1}{2}<s<\frac {3}{2}\), definitions of non-unique generalized and unique canonical co-normal derivatives 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.

##### MSC:

35J47 | Second-order elliptic systems |

35J57 | Boundary value problems for second-order elliptic systems |

35B65 | Smoothness and regularity of solutions to PDEs |