Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(0,L)\times E_a\subset \mathbb {R}^{d+1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <msub> <mi>E</mi> <mi>a</mi> </msub> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>d</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> denote the cylinder of length L, and base <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$E_a\subset \mathbb {R}^{d},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>E</mi> <mi>a</mi> </msub> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$E_a$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation> is an open ellipsoid with semi-axes <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$a=(a_1,...,a_d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>a</mi> <mi>d</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Let <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$w_{(0,L)\times E_a}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>w</mi> <mrow> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <msub> <mi>E</mi> <mi>a</mi> </msub> </mrow> </msub> </math></EquationSource> </InlineEquation> denote its torsion function. Heat equation tools are used to show that (i) if <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$E_a$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation> has small eccentricity and L is sufficiently large, then the maxima of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$|\nabla w_{(0,L)\times E_a}|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> </mrow> <msub> <mi>w</mi> <mrow> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <msub> <mi>E</mi> <mi>a</mi> </msub> </mrow> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are located at the centres (0, 0) and (L, 0) respectively, (ii) if <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$E_a$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation> has large eccentricity and L is sufficiently large, then the maxima are located on the lateral side of the cylinder.

On the Gradient of the Torsion Function for Elongated Cylinders

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2026年7月25日

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M. van den Berg

摘要

Let $(0,L)\times E_a\subset \mathbb {R}^{d+1}$ $(0, L) \times E_{a} \subset R^{d + 1}$ denote the cylinder of length L, and base $E_a\subset \mathbb {R}^{d},$ $E_{a} \subset R^{d},$ where $E_a$ $E_{a}$ is an open ellipsoid with semi-axes $a=(a_1,...,a_d)$ $a = (a_{1}, . . ., a_{d})$ . Let $w_{(0,L)\times E_a}$ $w_{(0, L) \times E_{a}}$ denote its torsion function. Heat equation tools are used to show that (i) if $E_a$ $E_{a}$ has small eccentricity and L is sufficiently large, then the maxima of $|\nabla w_{(0,L)\times E_a}|$ $| \nabla w_{(0, L) \times E_{a}} |$ are located at the centres (0, 0) and (L, 0) respectively, (ii) if $E_a$ $E_{a}$ has large eccentricity and L is sufficiently large, then the maxima are located on the lateral side of the cylinder.