We consider boundary value problems on manifolds with boundary where the boundary exhibits singularities of fibred cusp type. The simplest (unfibred) cusp is what is sometimes called an incomplete cusp, e.g. the complement of two touching circles in the plane. The fibred version includes the complement of two touching balls in $\mathbb{R}^n$. Our results also extend to geometries which are conformal to these incomplete cusps, for example fundamental domains of Fuchsian groups or uniformly fattened infinite cones in $\mathbb{R}^n$. For the Laplacian on such spaces, or more general elliptic operators $P$ whose structure relates well to the geometry, we study one of the basic objects of the theory of boundary value problems: the Calderon projector, which is essentially the projection to the set of boundary values (Cauchy data) of the homogeneous equation $Pu=0$. For a smooth compact manifold with boundary, it is classical that the Calderon projector is a pseudodifferential operator (PsiDO). In the case of the Laplacian, one can deduce from this that the Dirichlet-to-Neumann operator, which is fundamental to many spectral theoretic questions studied currently (like the Steklov spectrum), and also of interest in inverse problems, is a PsiDO also. We extend these results to the case where the boundary has fibred cusp singularities: both the Calderon projector and the Dirichlet-Neumann operator are in a PsiDO calculus adapted to the geometry, the so-called phi-calculus. This yields a precise description of their integral kernels near the singularities. In the talk, I will introduce the necessary background on the phi-calculus. This is joint work with K. Fritzsch and E. Schrohe.

Location: https://meet.google.com/you-qymk-ybu