In this paper, we study a kind of fully nonlinear equations of Monge-Ampère type, which can be applied to problems arising in optimal transport, geometric optics and conformal geometry. When the coefficient of the regular term a(x) has a positive lower bound, the purely interior Hessian estimates are proved for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. When the coefficient of the regular term is equal to zero, singular solutions can be constructed for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, while the purely interior Hessian estimates are obtained by proving Jacobi inequality and constructing proper auxiliary functions for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. As a byproduct, a new proof of the purely interior Hessian estimates for the two-dimensional standard Monge-Ampère equation is provided. For <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$a(x)\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, purely interior Hessian estimates for this kind of equations are still left open.

Purely interior Hessian estimates for a kind of Monge-Ampère equations

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发表日期：

2025年11月14日

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Minyang Cao,
Feida Jiang,
Wenhao Zhu

摘要

In this paper, we study a kind of fully nonlinear equations of Monge-Ampère type, which can be applied to problems arising in optimal transport, geometric optics and conformal geometry. When the coefficient of the regular term a(x) has a positive lower bound, the purely interior Hessian estimates are proved for $n\ge 2$ $n \geq 2$ . When the coefficient of the regular term is equal to zero, singular solutions can be constructed for $n\ge 3$ $n \geq 3$ , while the purely interior Hessian estimates are obtained by proving Jacobi inequality and constructing proper auxiliary functions for $n=2$ $n = 2$ . As a byproduct, a new proof of the purely interior Hessian estimates for the two-dimensional standard Monge-Ampère equation is provided. For $n=2$ $n = 2$ and $a(x)\ge 0$ $a (x) \geq 0$ , purely interior Hessian estimates for this kind of equations are still left open.