Let Ω be a bounded Lipschitz domain in Rn, n ≥ 3 with connected boundary. We study the Robin boundary condition ∂u/∂N + bu = f ∈ Lp(∂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.

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

Lanzani L.
;
Shen Z.
2005

Abstract

Let Ω be a bounded Lipschitz domain in Rn, n ≥ 3 with connected boundary. We study the Robin boundary condition ∂u/∂N + bu = f ∈ Lp(∂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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/873405
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