The average-distance problem with an Euler elastica penalization
Bibliographic record
Abstract
We consider the minimization of an average-distance functional defined on a two-dimensional domain \Omega with an Euler elastica penalization associated with \partial\Omega , the boundary of \Omega . The average distance is given by \int_{\Omega}\operatorname{dist}^p(x,\partial\Omega)\operatorname{d}x, where p\geq 1 is a given parameter and \operatorname{dist}(x,\partial\Omega) is the Hausdorff distance between \{x\} and \partial\Omega . The penalty term is a multiple of the Euler elastica (i.e., the Helfrich bending energy or the Willmore energy) of the boundary curve {\partial\Omega} , which is proportional to the integrated squared curvature defined on \partial\Omega , as given by \lambda\int_{\partial\Omega} \kappa_{\partial\Omega}^2 \operatorname{d}\mathcal{H}_{\llcorner\partial\Omega}^1, where \kappa_{\partial\Omega} denotes the (signed) curvature of \partial\Omega and \lambda>0 denotes a penalty constant. The domain \Omega is allowed to vary among compact, convex sets of \mathbb{R}^2 with Hausdorff dimension equal to two. Under no a priori assumptions on the regularity of the boundary \partial\Omega , we prove the existence of minimizers of E_{p,\lambda} . Moreover, we establish the C^{1,1} -regularity of its minimizers. An original construction of a suitable family of competitors plays a decisive role in proving the regularity.
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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) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.005 | 0.002 |
| 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".