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.
| Original language | English |
|---|---|
| Pages (from-to) | 91-109 |
| Number of pages | 19 |
| Journal | Communications in Partial Differential Equations |
| Volume | 29 |
| Issue number | 1-2 |
| DOIs | |
| State | Published - 2005 |
Bibliographical note
Funding Information:First author was supported by NSF Grant No. DMS-9800794. Second author was supported by NSF Grant No. DMS-9732894.
Funding
First author was supported by NSF Grant No. DMS-9800794. Second author was supported by NSF Grant No. DMS-9732894.
| Funders | Funder number |
|---|---|
| National Science Foundation (NSF) | DMS-9800794, DMS-9732894 |
Keywords
- Laplace's equation
- Lipschitz domains
- Robin boundary condition
ASJC Scopus subject areas
- Analysis
- Applied Mathematics