# Reference Question: Boundary Value Problem for Dirac Operator on Manifold with a non-smooth boundary

I am trying to find references for the following boundary value problem: Assume that $\Omega$ is a compact 3-dim spin manifold with Dirac operator $D$ such that the boundary consists of two smooth surfaces $\Sigma_1,\Sigma_2$ which meet orthogonally. For a spinor $\psi$, we consider the following boundary value problem

$$D(\psi)=\Psi \text{ in } \Omega \qquad P_{\pm}(\psi):=(id\pm i\gamma(N))\psi=P_{\pm}\eta \text{ on } \partial\Omega$$ for smooth spinors $\eta,\Psi$ on $\Omega$. Here, $N$ denotes the outward normal of the boundary and $\gamma$ the Clifford multiplication. Does anyone know a reference where existence, uniqueness and regularity of such boundary value problems has been considered? Thank you in advance.

There is a paper by R. Bartnik and P. Chrusciel that treats this question:

Bartnik, Robert, Chruściel, Piotr, Boundary value problems for Dirac-type equations, J. Reine Angew. Math. 579 (2005), 13–73.

And here is the arXiv link: https://arxiv.org/abs/math/0307278

• I'm a little bit confused, don't they assume the boundary to be smooth through out the paper? – user128470 Aug 5 '18 at 19:58
• At least according to the paper 'Index theory of Dirac operators on manifolds with corners up to codimension two', Advances and Applications, Vol. 151, 131-169 © 2004 Birkhiiuser Verlag, Basel/Switzerland, there does not seem to be any such theory as I just found out (c.f. section 5). – user128470 Aug 8 '18 at 12:01