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<p>The infinitesimal forms of the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(L_{p}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-Brunn–Minkowski inequalities for variational functionals, such as the <i>q</i>-capacity, the torsional rigidity, and the first eigenvalue of the Laplace operator, are investigated for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(p \ge 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. These formulations yield Poincaré-type inequalities related to these functionals. As an application, the <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(L_{p}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-Brunn–Minkowski inequalities for torsional rigidity with <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(0 \le p &lt; 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>p</mi> <mo>&lt;</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> are confirmed for small smooth perturbations of the unit ball.</p>

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The \(L_{p}\)-Brunn–Minkowski inequalities for variational functionals with \(0\le p<1\)

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  • 发表日期：

    2025年9月8日

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  • Jinrong Hu

摘要

The infinitesimal forms of the \(L_{p}\) L p -Brunn–Minkowski inequalities for variational functionals, such as the q-capacity, the torsional rigidity, and the first eigenvalue of the Laplace operator, are investigated for \(p \ge 0\) p ≥ 0 . These formulations yield Poincaré-type inequalities related to these functionals. As an application, the \(L_{p}\) L p -Brunn–Minkowski inequalities for torsional rigidity with \(0 \le p < 1\) 0 ≤ p < 1 are confirmed for small smooth perturbations of the unit ball.