Potential type operators and transmission problems for strongly elliptic second-order systems in Lipschitz domains

Abstract

We consider a strongly elliptic second-order system in a bounded n-dimensional domain Ω+ with Lipschitz boundary Γ, n ≥ 2. The smoothness assumptions on the coefficients are minimized. For convenience, we assume that the domain is contained in the standard torus \( \mathbb{T}^n \). In previous papers, we obtained results on the unique solvability of the Dirichlet and Neumann problems in the spaces Hpσ and Bpσ without use of surface potentials. In the present paper, using the approach proposed by Costabel and McLean, we define surface potentials and discuss their properties assuming that the Dirichlet and Neumann problems in Ω+ and the complementing domain Ω− are uniquely solvable. In particular, we prove the invertibility of the integral single layer operator and the hypersingular operator in Besov spaces on Γ. We describe some of their spectral properties as well as those of the corresponding transmission problems.