In this paper, we investigate the monotonicity of solutions to the nonlocal Monge-Ampère system <Equation ID="Equ44"> <EquationSource Format="TEX">$\begin{aligned} \left\{ \begin{array}{cc} D^\theta _s u(x)=f(u(x),v(x)),\ \ \ \ x \in \Omega , D^\theta _t v(x)=g(u(x),v(x)),\ \ \ \ x \in \Omega , \end{array}\right. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>D</mi> <mi>s</mi> <mi>θ</mi> </msubsup> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>v</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="4pt" /> <mspace width="4pt" /> <mspace width="4pt" /> <mspace width="4pt" /> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <msubsup> <mi>D</mi> <mi>t</mi> <mi>θ</mi> </msubsup> <mi>v</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>v</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="4pt" /> <mspace width="4pt" /> <mspace width="4pt" /> <mspace width="4pt" /> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$0<s, t<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\theta >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Omega \subseteq {\mathbb {R}}^{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊆</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> is a bounded domain which is convex in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$x_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>x</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>-direction or the whole space, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$D^\theta _{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>D</mi> <mi>α</mi> <mi>θ</mi> </msubsup> </math></EquationSource> </InlineEquation> is the nonlocal Monge-Ampère operator (<InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\alpha =s,t$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> </mrow> </math></EquationSource> </InlineEquation>), and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$f, g\in C^{1}({{\mathbb {R}}}^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>∈</mo> <msup> <mi>C</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We use the sliding method to prove any solution (u, v) of the system is strictly increasing in <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> with respect to <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$x_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>x</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> under some suitable conditions on f and g. The proof involves the idea that estimates the singular integrals along a sequence of approximate maximum points.

The monotonicity of bounded solutions for a nonlocal Monge-Ampère system in a bounded domain or the whole space

期刊论文
发表日期：

2025年12月22日

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Tingting Feng,
Zexin Zhang,
Zhitao Zhang

摘要

In this paper, we investigate the monotonicity of solutions to the nonlocal Monge-Ampère system $\begin{aligned} \left\{ \begin{array}{cc} D^\theta _s u(x)=f(u(x),v(x)),\ \ \ \ x \in \Omega , D^\theta _t v(x)=g(u(x),v(x)),\ \ \ \ x \in \Omega , \end{array}\right. \end{aligned}$ $\begin{matrix} \{\begin{matrix} D_{s}^{θ} u (x) = f (u (x), v (x)), x \in Ω, D_{t}^{θ} v (x) = g (u (x), v (x)), x \in Ω, \end{matrix}) \end{matrix}$ where $0<s, t<1$ $0 < s, t < 1$ , $\theta >0$ $θ > 0$ , $n\ge 2$ $n \geq 2$ , $\Omega \subseteq {\mathbb {R}}^{n}$ $Ω \subseteq R^{n}$ is a bounded domain which is convex in $x_n$ $x_{n}$ -direction or the whole space, $D^\theta _{\alpha }$ $D_{α}^{θ}$ is the nonlocal Monge-Ampère operator ( $\alpha =s,t$ $α = s, t$ ), and $f, g\in C^{1}({{\mathbb {R}}}^2)$ $f, g \in C^{1} (R^{2})$ . We use the sliding method to prove any solution (u, v) of the system is strictly increasing in $\Omega $ $Ω$ with respect to $x_n$ $x_{n}$ under some suitable conditions on f and g. The proof involves the idea that estimates the singular integrals along a sequence of approximate maximum points.