Bibliographic record
Abstract
We investigate the nonhomogeneous n-harmonic equation div|∇u| n-2 ∇u = f for U in the Sobolev space W 1,n (Ω), where f is a given function in the Zygmund class L log α L(Ω). In dimension n = 2 the solutions are continuous whenever f lies in the Hardy space Η 1 (Ω), so in particular, if f ∈ L log L(Ω). We show in higher dimensions that within the Zygmund classes the condition α > n - 1 is both necessary and sufficient for the solutions to be continuous. We also investigate continuity of the map f → ∇u, from L logα L(Ω) into L n log β L(Ω), for -1 < β < nα n-1 -1. These and other results of the present paper, though anticipated by simple examples, are in fact far from routine. Certainly, they are central in the p-harmonic theory.
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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 | 0.002 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.002 |
| 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".