While doing my research, I encountered the following problem as:
is there any regularity result for solutions to the Neumann problem for an elliptic PDE on a domain with piecewise smooth boundary?
For example, consider a domain whose boundary is the union of two smooth hypersurfaces with orthogonal normal field on their intersection. Let $f$ be the solution of the following Neumann problem:
\begin{equation}
\left\{\begin{array}{ll}
\mbox{div}(\nabla f)=g,& \mbox{ in }\Omega\\
f_{\nu}=g_1,&\mbox{ on }\Sigma_1\\
f_{\nu}=g_2,& \mbox{ on }\Sigma_2
\end{array}\right.
\end{equation}
where $\partial\Omega=\Sigma_1\cup\Sigma_2$, and $\Sigma_1$ and $\Sigma_2$ are smooth and perpendicular hypersurfaces. Is there any restriction on the data $g$, $g_1$, $g_2$ for the regularitiy? (we can even assume all of them are smooth.)
Any comments are welcomed.
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2$\begingroup$ 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). $\endgroup$– Daniele TampieriCommented Aug 20, 2021 at 4:36
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$\begingroup$ But we don't have a boundary even better than $C^1$. (while the non-smooth part measures zero.) $\endgroup$– yiminCommented Aug 20, 2021 at 5:17
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2$\begingroup$ 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. $\endgroup$– yiminCommented Aug 20, 2021 at 13:55
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1$\begingroup$ 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. :) $\endgroup$– Daniele TampieriCommented Aug 21, 2021 at 8:50
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1$\begingroup$ 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 $\endgroup$– yiminCommented Aug 21, 2021 at 12:26
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