In this paper, we investigate the Dirichlet problem for a class of mixed Hessian equations involving the modified Schouten tensor on Riemannian manifolds. In the non-degenerate case, we obtain a unique <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(k-1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-admissible solution removing the structural assumptions on the coefficients a(x) and b(x) that were required in previous works [<CitationRef CitationID="CR29">29</CitationRef>, <CitationRef CitationID="CR30">30</CitationRef>]. Then extending our analysis to the degenerate case, we derive the existence result in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(k-1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-admissible solutions under the regularity assumption that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(\alpha ^{l})^{\frac{1}{p_l}}\in C^{1,1}(\overline{M})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mi>l</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mi>l</mi> </msub> </mfrac> </msup> <mo>∈</mo> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mover> <mi>M</mi> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p_l\ge k-l.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mi>l</mi> </msub> <mo>≥</mo> <mi>k</mi> <mo>-</mo> <mi>l</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation>

The Dirichlet Problem for Degenerate Mixed Hessian Equations with the Modified Schouten Tensor on Riemannian Manifolds

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发表日期：

2026年2月16日

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Ni Xiang,
Yuni Xiong,
Botao Xu

摘要

In this paper, we investigate the Dirichlet problem for a class of mixed Hessian equations involving the modified Schouten tensor on Riemannian manifolds. In the non-degenerate case, we obtain a unique $(k-1)$ $(k - 1)$ -admissible solution removing the structural assumptions on the coefficients a(x) and b(x) that were required in previous works [29, 30]. Then extending our analysis to the degenerate case, we derive the existence result in $(k-1)$ $(k - 1)$ -admissible solutions under the regularity assumption that $(\alpha ^{l})^{\frac{1}{p_l}}\in C^{1,1}(\overline{M})$ ${(α^{l})}^{\frac{1}{p_{l}}} \in C^{1, 1} (\bar{M})$ , $p_l\ge k-l.$ $p_{l} \geq k - l .$