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| Natura: | Preprint |
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2023
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| Accesso online: | https://arxiv.org/abs/2302.08133 |
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| _version_ | 1866910453276344320 |
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| author | Monard, François Zou, Yuzhou |
| author_facet | Monard, François Zou, Yuzhou |
| contents | We consider a one-parameter family of degenerately elliptic operators $\cal{L}_γ$ 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 $γ\in \mathbb{R}$ in the sub-principal term, we prove that either the minimal operator is self-adjoint (case $|γ|\ge 1$), 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 $|γ|<1$). 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, object of interest in their own right, provide avenues to describe sufficient conditions for self-adjointness of extensions of $\cal{L}_{γ,min}$ that are parameterized in terms of boundary relations, and we formulate some corollaries to that effect. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_08133 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Boundary triples for a family of degenerate elliptic operators of Keldysh type Monard, François Zou, Yuzhou Analysis of PDEs Functional Analysis We consider a one-parameter family of degenerately elliptic operators $\cal{L}_γ$ 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 $γ\in \mathbb{R}$ in the sub-principal term, we prove that either the minimal operator is self-adjoint (case $|γ|\ge 1$), 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 $|γ|<1$). 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, object of interest in their own right, provide avenues to describe sufficient conditions for self-adjointness of extensions of $\cal{L}_{γ,min}$ that are parameterized in terms of boundary relations, and we formulate some corollaries to that effect. |
| title | Boundary triples for a family of degenerate elliptic operators of Keldysh type |
| topic | Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2302.08133 |