<p>We prove that measurable sets <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$E\subset \mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> with locally finite perimeter and zero <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$s$</EquationSource> <EquationSource Format="MATHML"><math> <mi>s</mi> </math></EquationSource> </InlineEquation>-mean curvature satisfy the surface density estimates: <Equation ID="Equ19"> <EquationSource Format="TEX">$\begin{aligned} \operatorname {Per} (E; B_R(x)) \ge CR^{n-1} \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mo>Per</mo> <mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo>;</mo> <msub> <mi>B</mi> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <mi>C</mi> <msup> <mi>R</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>for all <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$R>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$x\in \partial ^*E$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <msup> <mi>∂</mi> <mo>∗</mo> </msup> <mi>E</mi> </mrow> </math></EquationSource> </InlineEquation>. The constant <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$C$</EquationSource> <EquationSource Format="MATHML"><math> <mi>C</mi> </math></EquationSource> </InlineEquation> depends only on <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n$</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$s$</EquationSource> <EquationSource Format="MATHML"><math> <mi>s</mi> </math></EquationSource> </InlineEquation>, and remains bounded as <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$s\rightarrow 1^-$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo stretchy="false">→</mo> <msup> <mn>1</mn> <mo>-</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>. As an application, we prove that the fractional Sobolev inequality holds on the boundary of sets with zero <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$s$</EquationSource> <EquationSource Format="MATHML"><math> <mi>s</mi> </math></EquationSource> </InlineEquation>-mean curvature.</p>

Density estimates and the fractional Sobolev inequality for sets of zero $s$-mean curvature

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发表日期：

2025年11月11日

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Jack Thompson

摘要

We prove that measurable sets $E\subset \mathbb {R}^n$ $E \subset R^{n}$ with locally finite perimeter and zero $s$ $s$ -mean curvature satisfy the surface density estimates: $\begin{aligned} \operatorname {Per} (E; B_R(x)) \ge CR^{n-1} \end{aligned}$ $\begin{matrix} Per (E; B_{R} (x)) \geq C R^{n - 1} \end{matrix}$ for all $R>0$ $R > 0$ , $x\in \partial ^*E$ $x \in \partial^{*} E$ . The constant $C$ $C$ depends only on $n$ $n$ and $s$ $s$ , and remains bounded as $s\rightarrow 1^-$ $s \to 1^{-}$ . As an application, we prove that the fractional Sobolev inequality holds on the boundary of sets with zero $s$ $s$ -mean curvature.

Density estimates and the fractional Sobolev inequality for sets of zero \(s\)-mean curvature

摘要