A constructive approach to positive solutions of Δpu + f(u,∇u)≤0 on Riemannian manifolds
Bibliographic record
Abstract
Grigor'yan–Sun in [6] (with p = 2 ) and Sun in [10] (with p > 1 ) proved that if \sup_{r \gg 1}\mathrm{vol}(B(x_{0},r))r^{\frac{p\sigma }{p−\sigma −1}}(\mathrm{\ln }r)^{\frac{p−1}{p−\sigma −1}} < \infty then the only non-negative weak solution of \mathrm{\Delta }_{p}u + u^{\sigma } \leq 0 on a complete Riemannian manifold is identically 0; moreover, the powers of r and \mathrm{\ln }r are sharp. In this note, we present a constructive approach to the sharpness, which is flexible enough to treat the sharpness for \mathrm{\Delta }_{p}u + f(u,\mathrm{∇}u) \leq 0 . Our construction is based on a perturbation of the fundamental solution to the p -Laplace equation, and we believe that the ideas introduced here are applicable to other nonlinear differential inequalities on manifolds.
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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 | 0.008 | 0.011 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.006 | 0.002 |
| Science and technology studies | 0.002 | 0.007 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.003 | 0.005 |
| Research integrity | 0.003 | 0.006 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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".