<p>In this paper, we consider the Pogorelov estimates up to the flat boundary of convex solutions for Hessian quotient equations <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${{{S_{k}}(U[u])} \over {S_{l}(U[u])}}= {{{C}_{n}^{k}} \over {{C}_{n}^{l}}}{(n - 1)}^{k-l}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mfrac> <mrow> <mrow> <msub> <mi>S</mi> <mrow> <mi>k</mi> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">[</mo> <mi>u</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>S</mi> <mrow> <mi>l</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">[</mo> <mi>u</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow> <mfrac> <mrow> <msubsup> <mrow> <mi>C</mi> </mrow> <mrow> <mi>n</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msubsup> </mrow> <mrow> <msubsup> <mrow> <mi>C</mi> </mrow> <mrow> <mi>n</mi> </mrow> <mrow> <mi>l</mi> </mrow> </msubsup> </mrow> </mfrac> </mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mi>l</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$U[u] = (\Delta {u})I - D^{2}u, \, 0 \leq l < k \leq n$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>U</mi> <mo stretchy="false">[</mo> <mi>u</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mrow> <mi>u</mi> </mrow> <mo stretchy="false">)</mo> <mi>I</mi> <mo>−</mo> <msup> <mi>D</mi> <mrow> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mn>0</mn> <mo>≤</mo> <mi>l</mi> <mo><</mo> <mi>k</mi> <mo>≤</mo> <mi>n</mi> </math></EquationSource> </InlineEquation>. Furthermore, we obtain the Liouville theorem for such equation.</p>

Pogorelov Estimates and Liouville Theorem for Hessian Quotient Equations in Half Space

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发表日期：

2026年5月20日

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Xiaobiao Jia,
Wenfeng Yang,
Feifei Wang

摘要

In this paper, we consider the Pogorelov estimates up to the flat boundary of convex solutions for Hessian quotient equations ${{{S_{k}}(U[u])} \over {S_{l}(U[u])}}= {{{C}_{n}^{k}} \over {{C}_{n}^{l}}}{(n - 1)}^{k-l}$ $\frac{S_{k} (U [u])}{S_{l} (U [u])} = \frac{C_{n}^{k}}{C_{n}^{l}} {(n - 1)}^{k - l}$ , where $U[u] = (\Delta {u})I - D^{2}u, \, 0 \leq l < k \leq n$ $U [u] = (Δ u) I - D^{2} u, 0 \leq l < k \leq n$ . Furthermore, we obtain the Liouville theorem for such equation.