Well-posedness and convergence results for elliptic hemivariational inequalities
Bibliographic record
Abstract
We consider an elliptic hemivariational inequality in a real reflexive Banach space X which, under appropriate assumptions on the data, has a unique solution u ∈ X.We recall the concepts of wellposedness in the sense of Tykhonov and Levitin-Polyak for this inequality, and then we extend these concepts by introducing new well-posedness concepts, constructed with a larger set of approximating sequences.We also prove that, under additional assumptions, these new well-posedness concepts are optimal in the sense that all the sequences of elements of X which converge to the solution u are approximating sequences.This result, presented in Theorem 4.1, provides necessary and sufficient conditions for any sequence {u n } ⊂ X which guarantees that it converges to u and, therefore, it represents a convergence criterion to the solution of the hemivariational inequality.This criterion can be used in various applications.To provide an example, we illustrate its use in the study of a penalty method associated to an elliptic hemivariational inequality which describes the equilibrium of an elastic membrane in contact with a obstacle, the so-called foundation.
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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 | 0.004 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| 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".