Mixed problems in a Lipschitz domain for strongly elliptic second-order systems

Abstract

We consider mixed problems for strongly elliptic second-order systems in a bounded domain with Lipschitz boundary in the space ℝn. For such problems, equivalent equations on the boundary in the simplest L2-spaces Hs 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 Hps of Bessel potentials and Besov spaces Bps. 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.