In this paper, we investigate an <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3115_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$L_{p}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> Christoffel-Minkowski-type problem that prescribes a class of <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3115_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$L_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> geometric measures, which are mixtures of the k-th area measure and the q-th dual curvature measure. By establishing a gradient estimate, we obtain the existence of an even, smooth, strictly convex solution to this problem for <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3115_Article_IEq11.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="132" /> </InlineMediaObject> <EquationSource Format="TEX">$1 <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>q</mi> <mo>≤</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3115_Article_IEq12.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$1 \le k \le n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3115_Article_IEq13.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.

The $L_{p}$ dual Christoffel-Minkowski problem for $1 with \(1\le k\le n$

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2025年8月23日

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Carlos Cabezas-Moreno,
Jinrong Hu

摘要

In this paper, we investigate an $L_{p}$ $L_{p}$ Christoffel-Minkowski-type problem that prescribes a class of $L_p$ $L_{p}$ geometric measures, which are mixtures of the k-th area measure and the q-th dual curvature measure. By establishing a gradient estimate, we obtain the existence of an even, smooth, strictly convex solution to this problem for $1< p < q \le k + 1$ $1 < p < q \leq k + 1$ , where $1 \le k \le n$ $1 \leq k \leq n$ and $n \ge 1$ $n \geq 1$ .

The \(L_{p}\) dual Christoffel-Minkowski problem for \(1 with \(1\le k\le n\)

摘要