Elliptic equations with critical growth and a large set of boundary singularities
Why this work is in the frame
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Bibliographic record
Abstract
We solve variationally certain equations of stellar dynamics of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus sigma-summation Underscript i Endscripts partial-differential Subscript i i Baseline u left-parenthesis x right-parenthesis equals StartFraction StartAbsoluteValue u EndAbsoluteValue Superscript p minus 2 Baseline u left-parenthesis x right-parenthesis Over dist left-parenthesis x comma script upper A right-parenthesis Superscript s Baseline EndFraction"> <mml:semantics> <mml:mrow> <mml:mo> − </mml:mo> <mml:munder> <mml:mo> ∑ </mml:mo> <mml:mi>i</mml:mi> </mml:munder> <mml:msub> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mi>i</mml:mi> </mml:mrow> </mml:msub> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>u</mml:mi> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> <mml:mo> − </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>dist</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> </mml:mrow> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>s</mml:mi> </mml:msup> </mml:mrow> </mml:mfrac> </mml:mrow> <mml:annotation encoding="application/x-tex">-\sum _i\partial _{ii} u(x) =\frac {|u|^{p-2}u(x)}{\textrm {dist} (x,{\mathcal A} )^s}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a domain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathcal A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a proper linear subspace of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Existence problems are related to the question of attainability of the best constant in the following inequality due to Maz’ya (1985): <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than mu Subscript s comma script upper P Baseline left-parenthesis normal upper Omega right-parenthesis equals inf left-brace integral Underscript normal upper Omega Endscripts StartAbsoluteValue nabla u EndAbsoluteValue squared d x vertical-bar u element-of upper H Subscript 1 comma 0 Superscript 2 Baseline left-parenthesis normal upper Omega right-parenthesis normal a normal n normal d integral Underscript normal upper Omega Endscripts StartFraction StartAbsoluteValue u left-parenthesis x right-parenthesis EndAbsoluteValue Superscript 2 Super Superscript star Superscript left-parenthesis s right-parenthesis Baseline Over StartAbsoluteValue pi left-parenthesis x right-parenthesis EndAbsoluteValue Superscript s Baseline EndFraction d x equals 1 right-brace comma"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:msub> <mml:mi> μ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo movablelimits="true" form="prefix">inf</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:msub> <mml:mo> ∫ </mml:mo>
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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| 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