一类Keldysh型退化椭圆算子的边界三元组
Boundary triples for a family of degenerate elliptic operators of Keldysh type
AI总结：
该研究针对闭圆盘上与测地线X射线变换相关的单参数Keldysh型退化椭圆算子，按次主项参数γ的取值分类证明极小算子自伴性，构造|γ|<1时的迹映射与Sobolev尺度，并用边界三元组解释相关结果，给出算子扩张自伴性的充分条件推论。
摘要：
我们研究闭圆盘$\u2139$上的一类单参数Keldysh（或称Kimura）型退化椭圆算子$\u2112_γ$，该算子曾出现在作者与Mishra此前的研究[Mishra等人，Inverse Problems (2022)]中，与测地线X射线变换相关。根据次主项中常数$γ∈ℝ$的取值，我们证明：当$|γ|≥1$时，极小算子是自伴的；当$|γ|<1$时，可以构造合适的迹映射以及（$\u2139$和$\u211D^1=∂\u2139$上的）Sobolev尺度，用以刻画映射性质、Dirichlet-to-Neumann映射，并推广Green恒等式。后者可通过极大算子的边界三元组，或其一个特殊限制的广义边界三元组来重新解释。这些概念本身就具有研究价值，为描述由边界关系参数化的$\u2112_{γ,min}$扩张的自伴性充分条件提供了途径，我们还就此给出了若干推论。
英文摘要：
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.
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