Remarks on strongly elliptic systems in Lipschitz domains

Abstract

We present some remarks to the general theory of strongly elliptic second-order systems in bounded Lipschitz domains. The most important remarks are related to the use of the “Weyl decomposition” of the solution space. In particular, we suggest a simplified approach to the unique choice of the right-hand side of the system and the conormal derivative in the Neumann problem and obtain two-sided a priori estimates for the solutions. We consider the transmission problem for two systems in domains with a common Lipschitz boundary without the assumption that the coefficients do not have jumps on that boundary. We construct examples of strongly elliptic second-order systems for which the Neumann problem is not Fredholm.