On the Robin Boundary Condition for Laplace's Equation in Lipschitz Domains

Abstract

Let Ω be a bounded Lipschitz domain in ℝ n , n ≥ 3 with connected boundary. We study the Robin boundary condition ∂u/∂N + bu = f ∈ L p (∂Ω) on ∂Ω for Laplace's equation Δu = 0 in Ω, where b is a non-negative function on ∂Ω. For 1 < p < 2 + ϵ, under suitable compatibility conditions on b, we obtain existence and uniqueness results with non-tangential maximal function estimate ‖(∇u)*‖ p ≤ C‖f‖ p , as well as a pointwise estimate for the associated Robin function. Moreover, the solution u is represented by a single layer potential.