Mixed boundary value problem of Laplace equation in a bounded Lipschitz domain

We study the existence and uniqueness of the mixed boundary value problem for Laplace equation in a bounded Lipschitz domain Ω ⊂ Rn, n 3. Let the boundary ∂Ω of Ω be decomposed by ∂Ω = Γ = Γ1 ∪ Γ 2 = Γ 1 ∪ Γ2, Γ1 ∩ Γ2 = ∅. We will show that if the Neumann data ψ is in H−1 2 (Γ2) and the Dirichlet data f is in H 12 (Γ1), then the mixed boundary value problem has a unique solution and the solution is represented by potentials. © 2007 Elsevier Inc. All rights reserved