Hardy–Sobolev critical elliptic equations with boundary singularities
Bibliographic record
Abstract
Unlike the non-singular case s=0 , or the case when 0 belongs to the interior of a domain Ω in ℝ^n ( n⩾3 ), we show that the value and the attainability of the best Hardy–Sobolev constant on a smooth domain Ω , \mu _{s}(\Omega ): = \inf \left\{\int \limits_{\Omega }\left|\nabla u\right|^{2}dx;u∊H_{0}^{1}(\Omega )\text{ and }\int \limits_{\Omega }\frac{\left|u\right|^{2*(s)}}{\left|x\right|^{s}} = 1\right\} when 0 , 2^∗(s)=\frac{2(n - s)}{n - 2} , and when 0 is on the boundary ∂Ω are closely related to the properties of the curvature of ∂Ω at 0. These conditions on the curvature are also relevant to the study of elliptic partial differential equations with singular potentials of the form: - \Delta u = \frac{u^{p - 1}}{\left|x\right|^{s}} + f(x,u)\text{ in }\Omega \subset ℝ^{n}, where f is a lower order perturbative term at infinity and f(x,0)=0 . We show that the positivity of the sectional curvature at 0 is relevant when dealing with Dirichlet boundary conditions, while the Neumann problems seem to require the positivity of the mean curvature at 0. Résumé Contrairement au cas non-singulier s=0 , ou au cas d’une singularité à l’intérieur d’ un domaine Ω de ℝ^n ( n⩾3 ), on montre que la valeur de la meilleure constante dans l’inégalité de Hardy–Sobolev sur un domaine régulier, \mu _{s}(\Omega ): = \inf \left\{\int \limits_{\Omega }\left|\nabla u\right|^{2}dx;u∊H_{0}^{1}(\Omega )\text{ et }\int \limits_{\Omega }\frac{\left|u\right|^{2*(s)}}{\left|x\right|^{s}} = 1\right\} quand 0 , 2^∗(s)=\frac{2(n - s)}{n - 2} , et quand 0 appartient à la frontière, est étroitement liée aux propriétés de la courbure de ∂Ω en 0. Ces mêmes conditions sur la courbure sont aussi pertinentes pour l’existence de solutions d’équations à potentiel singulier de la forme : - \Delta u = \frac{u^{p - 1}}{\left|x\right|^{s}} + f(x,u)\text{ in }\Omega \subset ℝ^{n}, où f est une perturbation d’ordre inférieur à l’infini et f(x,0)=0 . On montre que la positivité de la courbure sectionelle est suffisante pour l’existence de solutions des problèmes avec conditions de Dirichlet au bord, tandis que pour les problèmes de Neumann, c’est la positivité de la coubure moyenne qui compte.
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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 | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.002 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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".