Abstract
Given p > 2 and a bounded Lipschitz domain Ω, we obtain a necessary and sufficient condition for the solvability of the Neumann problem for Laplace's equation Δ u = 0 in Ω with boundary data in Lp (∂ Ω). As a result, we show that the Lp Neumann problem on convex domains in Rd is solvable for 1 < p < ∞ if d = 2, for 1 < p < 4 if d = 3, and for 1 < p < 3 + ε if d ≥ 4.
| Original language | English |
|---|---|
| Pages (from-to) | 1817-1830 |
| Number of pages | 14 |
| Journal | Journal of Functional Analysis |
| Volume | 255 |
| Issue number | 7 |
| DOIs | |
| State | Published - Oct 1 2008 |
Bibliographical note
Funding Information:The authors are supported in part by the NSF (DMS-0500257). Corresponding author. E-mail address: [email protected] (Z. Shen).
Funding
The authors are supported in part by the NSF (DMS-0500257). Corresponding author. E-mail address: [email protected] (Z. Shen).
| Funders | Funder number |
|---|---|
| U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China | DMS-0500257 |
| U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China |
Keywords
- Convex domain
- Lipschitz domain
- Neumann problem
ASJC Scopus subject areas
- Analysis