Mixed Problems in a Lipschitz Domain for Strongly Elliptic Second-Order Systems

We consider mixed problems for strongly elliptic second-order systems in a bounded domain with Lipschitz boundary in the space $\mathbb{R}^n$. For such problems, equivalent equations on the boundary in the simplest $L_2$-spaces $H^s$ of Sobolev type are derived, which permits one to represent the solutions via surface potentials. We prove a result on the regularity of solutions in the slightly more general spaces $H^s_p$ of Bessel potentials and Besov spaces $B^s_p$. Problems with spectral parameter in the system or in the condition on a part of the boundary are considered, and the spectral properties of the corresponding operators, including the eigenvalue asymptotics, are discussed.