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.
Lanzani, L., Shen, Z. (2005). On the Robin Boundary Condition for Laplace's Equation in Lipschitz Domains. COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, 29(1-2), 91-109 [10.1081/pde-120028845].
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
