Boundary triples for a family of degenerate elliptic operators of Keldysh type

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Abstract

We consider a one-parameter family of degenerately elliptic operators $\cal{L}_\gamma$ on the closed disk $\mathbb{D}$, of Keldysh (or Kimura) type, which appears in prior work [Mishra et al., Inverse Problems (2022)] by the authors and Mishra, related to the geodesic X-ray transform. Depending on the value of a constant $\gamma\in \mathbb{R}$ in the sub-principal term, we prove that either the minimal operator is self-adjoint (case $|\gamma|\ge 1$), or that one may construct appropriate trace maps and Sobolev scales (on $\mathbb{D}$ and $\mathbb{S}^1=\partial\mathbb{D}$) on which to formulate mapping properties, Dirichlet-to-Neumann maps, and extend Green's identities (case $|\gamma|<1$). The latter can be reinterpreted in terms of a boundary triple for the maximal operator, or a generalized boundary triple for a distinguished restriction of it. The latter concepts, object of interest in their own right, provide avenues to describe sufficient conditions for self-adjointness of extensions of $\cal{L}_{\gamma,min}$ that are parameterized in terms of boundary relations, and we formulate some corollaries to that effect.