On the Neumann Problem for the Schrödinger Equations with Singular Potentials in Lipschitz Domains
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Bibliographic record
Abstract
Abstract We consider the Neumann problem for the Schrödinger equations –Δ u + Vu = 0, with singular nonnegative potentials V belonging to the reverse Hölder class ℬ n , in a connected Lipschitz domain Ω R n . Given boundary data g in H p or L p for 1 – ɛ < p ≤ 2, where 0 < ɛ < , it is shown that there is a unique solution, u , that solves the Neumann problem for the given data and such that the nontangential maximal function of ▽ u is in L p (∂Ω). Moreover, the uniform estimates are found.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it