The singular extremal solutions of the bi-Laplacian with exponential nonlinearity
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Bibliographic record
Abstract
Consider the problem <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row StartLayout Enlarged left-brace 1st Row 1st Column normal upper Delta squared u equals lamda e Superscript u Baseline 2nd Column a m p semicolon in upper B comma 2nd Row 1st Column u equals StartFraction partial-differential u Over partial-differential n EndFraction equals 0 2nd Column a m p semicolon on partial-differential upper B comma EndLayout EndLayout"> <mml:semantics> <mml:mtable columnalign="right center left" rowspacing="3pt" columnspacing="0 thickmathspace" side="left" displaystyle="true"> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable columnalign="left left" rowspacing="4pt" columnspacing="1em"> <mml:mtr> <mml:mtd> <mml:msup> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi> λ </mml:mi> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>u</mml:mi> </mml:mrow> </mml:msup> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mtext>in </mml:mtext> <mml:mi>B</mml:mi> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mi>u</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mi>n</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mtext>on </mml:mtext> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mi>B</mml:mi> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:mo fence="true" stretchy="true" symmetric="true"/> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> <mml:annotation encoding="application/x-tex">\begin{eqnarray*} \left \{ \begin {array}{ll} \Delta ^2 u= \lambda e^{u} &\text {in } B,\\ u=\frac {\partial u}{\partial n}=0 &\text {on }\partial B, \end{array} \right . \end{eqnarray*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the unit ball in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript upper N"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </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> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda"> <mml:semantics> <mml:mi> λ </mml:mi> <mml:annotation encoding="application/x-tex">\lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a parameter. Unlike the Gelfand problem the natural candidate <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u equals minus 4 ln left-parenthesis StartAbsoluteValue x EndAbsoluteValue right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mo> − </mml:mo> <mml:mn>4</mml:mn> <mml:mi>ln</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">u=-4\ln (|x|)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , for the extremal solution, does not satisfy the boundary conditions, and hence showing the singular nature of the extremal solution in large dimensions close to the critical dimension is challenging. Recently a computer-assisted proof was used to show that the extremal solution is singular in dimensions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="13 less-than-or-equal-to upper N less-than-or-equal-to 31"> <mml:semantics> <mml:mrow> <mml:mn>13</mml:mn> <mml:mo> ≤ </mml:mo> <mml:mi>N</mml:mi>
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 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