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

François Monard and Yuzhou (Joey) Zou

Vol. 6 (2024), No. 2, 541–580

We consider a one-parameter family of degenerately elliptic operators γ on the closed disk 𝔻, of Keldysh (or Kimura) type, which appears in prior work by the authors and Mishra (2023), related to the geodesic X-ray transform. Depending on the value of a constant γ in the subprincipal term, we prove that either the minimal operator is self-adjoint (case |γ| 1), or that one may construct appropriate trace maps and Sobolev scales (on 𝔻 and 𝕊1 = 𝔻) on which to formulate mapping properties, Dirichlet-to-Neumann maps, and extend Green’s identities (case |γ| < 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, objects of interest in their own right, provide avenues to describe sufficient conditions for self-adjointness of extensions of γ,min that are parametrized in terms of boundary relations, and we formulate some corollaries to that effect.

Keldysh operators, Kimura operators, boundary triple, generalized boundary triple, X-ray transform, degenerately elliptic operators
Mathematical Subject Classification
Primary: 35J67, 35J70, 47B25, 47F10
Secondary: 26B20
Received: 13 July 2023
Revised: 5 December 2023
Accepted: 18 January 2024
Published: 16 May 2024
François Monard
Department of Mathematics
University of California Santa Cruz
Santa Cruz, CA
United States
Yuzhou (Joey) Zou
Northwestern University
Department of Mathematics
Evanston, IL
United States