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.
%0 Journal Article
%1 noauthororeditor
%A Lanzani, Loredana
%A Shen, Zhongwei
%D 2004
%J Communications in Partial Differential Equations
%K 35j05-laplacian-operator-helmholtz-poisson-equation
%N 1-2
%P 91-109
%R 10.1081/PDE-120028845
%T On the Robin Boundary Condition for Laplace's Equation in Lipschitz Domains
%U https://www.tandfonline.com/doi/abs/10.1081/PDE-120028845
%V 29
%X 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.
@article{noauthororeditor,
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.
},
added-at = {2021-04-12T09:08:41.000+0200},
author = {Lanzani, Loredana and Shen, Zhongwei},
biburl = {https://www.bibsonomy.org/bibtex/2f5ec2a3e6b4835012b54a5e632d1ac36/gdmcbain},
doi = {10.1081/PDE-120028845},
interhash = {48633f62c186bbecb26676b2e701ca57},
intrahash = {f5ec2a3e6b4835012b54a5e632d1ac36},
journal = {Communications in Partial Differential Equations},
keywords = {35j05-laplacian-operator-helmholtz-poisson-equation},
number = {1-2},
pages = {91-109},
timestamp = {2021-04-12T09:08:41.000+0200},
title = {On the Robin Boundary Condition for Laplace's Equation in Lipschitz Domains
},
url = {https://www.tandfonline.com/doi/abs/10.1081/PDE-120028845},
volume = 29,
year = 2004
}