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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.

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  • $\begingroup$ Can you elaborate a bit on the fact that the boundary consists of two surfaces meeting orthogonally? That is not standard. What would be an example of such a thing in lower dimension say $2$ instead of $3$? $\endgroup$
    – Overflowian
    Dec 16, 2021 at 11:25
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    $\begingroup$ @Overflowian A square, for instance, or a well-chosen spherical triangle. This is called a manifold with codimension-2 corners (and the orthogonality condition is a condition on the metric near these corner strata). $\endgroup$
    – mme
    Sep 12, 2022 at 15:18

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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

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    $\begingroup$ I'm a little bit confused, don't they assume the boundary to be smooth through out the paper? $\endgroup$
    – user128470
    Aug 5, 2018 at 19:58
  • $\begingroup$ 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). $\endgroup$
    – user128470
    Aug 8, 2018 at 12:01

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