On the dimension of $p$-harmonic measure in space
Bibliographic record
Abstract
Let \Omega\subset\mathbb {R}^{n} , n\geq 3 , and let p , 1 < p < \infty , p \not = 2 , be given. In this paper we study the dimension of p -harmonic measures that arise from non-negative solutions to the p -Laplace equation, vanishing on a portion of \partial\Omega , in the setting of \delta -Reifenberg flat domains. We prove, for p \geq n , that there exists \tilde\delta=\tilde\delta(p,n)>0 small such that if \Omega is a \delta -Reifenberg flat domain with \delta<\tilde\delta , then p -harmonic measure is concentrated on a set of \sigma -finite H^{n-1} -measure. We prove, for p \geq n , that for sufficiently flat Wolff snowflakes the Hausdorff dimension of p -harmonic measure is always less than n-1 . We also prove that if 2 , then there exist Wolff snowflakes such that the Hausdorff dimension of p -harmonic measure is less than n-1 , while if 1 , then there exist Wolff snowflakes such that the Hausdorff dimension of p -harmonic measure is larger than n-1 . Furthermore, perturbing off the case p = 2, we derive estimates when p is near 2 for the Hausdorff dimension of p -harmonic measure.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".