In this paper, we consider the Pogorelov estimate up to the boundary and the Liouville type theorem for parabolic k-Hessian equations in half spaces. We prove that any k-convex-monotone solutions u of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1909_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="133" /> </InlineMediaObject> <EquationSource Format="TEX">$-u_t\sigma _k(D^2u)=C_n^k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <msub> <mi>u</mi> <mi>t</mi> </msub> <msub> <mi>σ</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>D</mi> <mn>2</mn> </msup> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msubsup> <mi>C</mi> <mi>n</mi> <mi>k</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation> with boundary value <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1909_Article_IEq2.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="174" /> </InlineMediaObject> <EquationSource Format="TEX">$u(x',x_n,t)=-t+\frac{1}{2}|x^{\prime }|^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo>-</mo> <mi>t</mi> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mrow> <mo stretchy="false">|</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1909_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">$\{x_n=0\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo>=</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> must be a quadratic polynomial, provided that u satisfies some proper conditions.

The Pogorelov estimate and Liouville theorem for parabolic k-Hessian equations in half spaces

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发表日期：

2025年6月16日

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Xiaobiao Jia,
Shanshan Ma

摘要

In this paper, we consider the Pogorelov estimate up to the boundary and the Liouville type theorem for parabolic k-Hessian equations in half spaces. We prove that any k-convex-monotone solutions u of $-u_t\sigma _k(D^2u)=C_n^k$ $- u_{t} σ_{k} (D^{2} u) = C_{n}^{k}$ with boundary value $u(x',x_n,t)=-t+\frac{1}{2}|x^{\prime }|^2$ $u (x^{'}, x_{n}, t) = - t + \frac{1}{2} {| x^{'} |}^{2}$ on $\{x_n=0\}$ ${x_{n} = 0}$ must be a quadratic polynomial, provided that u satisfies some proper conditions.