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

Abstract

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