Timeline for The regularity of solutions to the Neumann problem for an elliptic PDE on a domain with piecewise smooth boundary
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 21, 2021 at 12:26 | comment | added | yimin | In Chapter 1, they assume the boundary to be smooth, but from the descriptions at the beginning of Chapter 8, the model of the boundary containing a smooth submanifold, at the point of the submanifold, has a neighborhood diffeomorphic to a production of a cone with a Euclidean space, which seems to match with my model | |
| Aug 21, 2021 at 8:50 | comment | added | Daniele Tampieri | Be careful, since Nazarov and Plamenevsky consider "smooth $=C^\infty$" (op. cit. chapter 1, §1.1 p. 4, where they give a brief description of the notation). The reason for developing the theory assuming $\partial\Omega\in C^2$ or at least a to be a Lyapunov manifold ($C^{1,\alpha}$, $0<\alpha\le 1$) is possibly due to the fact that you have at least an "estimate" on the direction of the outer normal to $\partial\Omega$. However, Nazarov and Plamenevsky's monograph is a nice work, so have an app reading experience. :) | |
| Aug 20, 2021 at 13:55 | comment | added | yimin | I guess I have found the answer in a book named"Elliptic Problems in Domains with Piecewise Smooth Boundaries" written by Sergey A. Nazarov, Boris A. Plamenevsky for my question. The boundary model in Chapter 8 is exactly the case I need. | |
| Aug 20, 2021 at 5:17 | comment | added | yimin | But we don't have a boundary even better than $C^1$. (while the non-smooth part measures zero.) | |
| Aug 20, 2021 at 4:36 | comment | added | Daniele Tampieri | For the standard smooth case this Q&A is relevant: the piecewise smooth boundary case seems to be tractable with the same techniques used for the globally smooth (read $\partial\Omega\in C^{k+2}$, $k\in\Bbb N$ case). | |
| Aug 20, 2021 at 4:32 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor grammar improvements and formatting
|
| Aug 20, 2021 at 4:05 | review | First posts | |||
| Aug 20, 2021 at 6:05 | |||||
| Aug 20, 2021 at 4:01 | history | asked | yimin | CC BY-SA 4.0 |