Noncoercive elliptic bilateral variational inequalities in the homogeneous Sobolev space $D^{1,p}(R^N)$
Bibliographic record
Abstract
In this paper, we prove an existence result for a quasilinear elliptic variational inequality of the form u ∈ K ⊂ V : 0 ∈ -∆ p u+aF(ua.e. in R N }, where ∆ p is the p-Laplacian, the underlying solution space V is the homogeneous Sobolev space (also called Beppo-Levi space) V = D 1,p (R N ) with 1 < p < N, and I K : V → R ∪ {+∞} is the indicator functional corresponding to K with its subdifferential ∂ I K .The lower order Nemytskij operator F is generated by a Carathéodory function f : R N ×R → R, and the measurable and bounded coefficient a is supposed to decay like |x| -(N+α) at infinity.The growth conditions that we impose on f are such that the operator -∆ p + aF : V → V * , in general, is not coercive with respect to K which prevents us from applying standard existence results.Another difficulty, which arises due to the lack of compact embedding of V into L q (R N ) spaces, needs to be overcome in an appropriate way.Without assuming additional assumptions such as the existence of sub-and supersolutions, we are able not only to prove the existence of solutions, but also show the compactness of the set of all solutions in V .Finally, an extension of the theory is established, which allows us to deal with noncoercive bilateral variational-hemivariational inequalities in R N .The proof of our main existence result is based on a modified penalty approach.
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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 | 0.003 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.006 | 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".