<p>Diameter estimates of solutions to 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> Aleksandrov problem for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$-1<p\le 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo>≤</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> are established in this paper. As an application, we give a different proof of the existence of solutions to the even <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$L_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> Aleksandrov problem for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$-1<p<0,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> which was obtained by Mui (Adv Math 408:108573, 2022) based on the variational method. We also find that <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$p=-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> is a critical value for this problem by constructing some examples showing that the existence (resp. uniqueness) may fail for general even measures when <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$p=-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> (resp. <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$p<-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo><</mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>).</p>

Diameter estimates of the $L_p$ Aleksandrov problem for \(-1

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2026年3月10日

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Yibin Feng,
Shengnan Hu,
Yuanyuan Li,
Honglin Lv

摘要

Diameter estimates of solutions to the $L_p$ $L_{p}$ Aleksandrov problem for $-1<p\le 0$ $- 1 < p \leq 0$ are established in this paper. As an application, we give a different proof of the existence of solutions to the even $L_p$ $L_{p}$ Aleksandrov problem for $-1<p<0,$ $- 1 < p < 0,$ which was obtained by Mui (Adv Math 408:108573, 2022) based on the variational method. We also find that $p=-1$ $p = - 1$ is a critical value for this problem by constructing some examples showing that the existence (resp. uniqueness) may fail for general even measures when $p=-1$ $p = - 1$ (resp. $p<-1$ $p < - 1$ ).

Diameter estimates of the \(L_p\) Aleksandrov problem for \(-1

摘要