Maximizers for the singular Trudinger-Moser inequalities in the subcritical cases
Why this work is in the frame
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Bibliographic record
Abstract
The main purpose of this note is to study the existence of extremal functions for the singular Trudinger-Moser inequalities in the subcritical cases. More precisely, let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">N\geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than beta greater-than upper N comma 0 greater-than a comma b"> <mml:semantics> <mml:mrow> <mml:mtext> </mml:mtext> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi> β </mml:mi> <mml:mo>></mml:mo> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> <mml:mtext> </mml:mtext> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mtext> </mml:mtext> <mml:mi>b</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">~0>\beta >N,~0>a,~b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and denote <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column upper T upper M Subscript a comma b comma beta Baseline left-parenthesis alpha right-parenthesis 2nd Column a m p semicolon equals sup Underscript double-vertical-bar nabla u double-vertical-bar Subscript upper N Superscript a Baseline plus double-vertical-bar u double-vertical-bar Subscript upper N Superscript b Baseline less-than-or-equal-to 1 Endscripts integral Underscript double-struck upper R Superscript upper N Baseline Endscripts phi Subscript upper N Baseline left-parenthesis alpha left-parenthesis 1 minus StartFraction beta Over upper N EndFraction right-parenthesis StartAbsoluteValue u EndAbsoluteValue Superscript StartFraction upper N Over upper N minus 1 EndFraction Baseline right-parenthesis StartFraction d x Over StartAbsoluteValue x EndAbsoluteValue Superscript beta Baseline EndFraction comma 2nd Row 1st Column phi Subscript upper N Baseline left-parenthesis t right-parenthesis 2nd Column a m p semicolon equals e Superscript t Baseline minus sigma-summation Underscript j equals 0 Overscript upper N minus 2 Endscripts StartFraction t Superscript j Baseline Over j factorial EndFraction period EndLayout"> <mml:semantics> <mml:mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" side="left" displaystyle="true"> <mml:mtr> <mml:mtd> <mml:mi>T</mml:mi> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>,</mml:mo> <mml:mi> β </mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi> α </mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mo>=</mml:mo> <mml:munder> <mml:mo movablelimits="true" form="prefix">sup</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msubsup> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mi mathvariant="normal"> ∇ </mml:mi> <mml:mi>u</mml:mi> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>a</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mi>u</mml:mi> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>b</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo> ≤ </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:munder> <mml:msub> <mml:mo> ∫ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:msub> <mml:msub> <mml:mi> ϕ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi> α </mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo> − </mml:mo> <mml:mfrac> <mml:mi> β </mml:mi> <mml:mi>N</mml:mi> </mml:mfrac> <mml:mo>)</mml:mo> </mm
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.003 | 0.010 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.006 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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