# 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.

• 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 $3$? Commented Dec 16, 2021 at 11:25
• @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).
– mme
Commented Sep 12, 2022 at 15:18