Sharp bounds for higher Steklov–Dirichlet eigenvalues on domains with spherical holes
Bibliographic record
Abstract
Abstract We consider a mixed Steklov–Dirichlet eigenvalue problem on a smooth bounded domain having a spherical hole. In this article, we take Dirichlet condition on the boundary of the spherical hole and Steklov condition on the other boundary component/s. Under certain symmetry assumptions on multiconnected domains in $\mathbb {R}^{n}$ having a spherical hole, we obtain isoperimetric inequalities for the k -th Steklov–Dirichlet eigenvalues for each $k \in \{2, 3, \dots , n+1\}$ . We provide examples to emphasise the fact that the symmetry assumptions, on the family of domains considered, are crucial. We also extend Theorem 3.1 of “Gavitone et al. (2023), An isoperimetric inequality for the first Steklov–Dirichlet Laplacian eigenvalue of convex sets with a spherical hole, Pacific Journal of Mathematics , 320(2): 241–259” not only from Euclidean domains to domains in space forms but also from convex domains to star-shaped domains. In particular, we obtain sharp lower and upper bounds for the first Steklov–Dirichlet eigenvalue on the family of all bounded star-shaped domains on the hemisphere as well as on the hyperbolic space.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.016 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.004 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".