We characterize bounded simply connected planar W1,p-extension domains for 1 < p < 2 as those bounded simply connected domains Ω ⊂ ℝ2 for which any two points z1, z2 ∈ ℝ2 Ω can be connected with a curve γ ⊂ ℝ2 Ω satisfying<Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11425_2023_2339_Article_Equa.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="337" /> </MediaObject> <EquationSource Format="TEX">$\int_{\gamma}\text{dist}(z,\partial\Omega)^{1-p} ds(z)\leqslant C(\Omega,p)\mid z_{1}-z_{2}\mid^{2-p}.$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mo>∫</mo> <mrow> <mi>γ</mi> </mrow> </msub> <mtext>dist</mtext> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Ω</mi> <msup> <mo stretchy="false">)</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </msup> <mi>d</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>⩽</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <msub> <mi>z</mi> <mrow> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow> <mn>2</mn> </mrow> </msub> <msup> <mo>∣</mo> <mrow> <mn>2</mn> <mo>−</mo> <mi>p</mi> </mrow> </msup> <mo>.</mo> </math></EquationSource> </Equation>By combining earlier results, we obtain the following duality result: a Jordan domain Ω ⊂ ℝ2 is a W1,p-extension domain, 1 < p < ∞ if and only if the complementary domain <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11425_2023_2339_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb{R}^{2}\backslash\bar{\Omega}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mi class="MJX-variant" mathvariant="normal">∖</mi> <mrow> <mover> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> is a W1,p/(p−1)-extension domain.

A geometric characterization of planar Sobolev extension domains

期刊论文
发表日期：

2025年2月27日

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Pekka Koskela,
Tapio Rajala,
Yi Ru-Ya Zhang

摘要

We characterize bounded simply connected planar W^1,p-extension domains for 1 < p < 2 as those bounded simply connected domains Ω ⊂ ℝ² for which any two points z₁, z₂ ∈ ℝ² Ω can be connected with a curve γ ⊂ ℝ² Ω satisfying $\int_{\gamma}\text{dist}(z,\partial\Omega)^{1-p} ds(z)\leqslant C(\Omega,p)\mid z_{1}-z_{2}\mid^{2-p}.$ $\int_{γ} dist (z, \partial Ω)^{1 - p} d s (z) ⩽ C (Ω, p) ∣ z_{1} - z_{2} ∣^{2 - p} .$

By combining earlier results, we obtain the following duality result: a Jordan domain Ω ⊂ ℝ² is a W^1,p-extension domain, 1 < p < ∞ if and only if the complementary domain $\mathbb{R}^{2}\backslash\bar{\Omega}$ $R^{2} ∖ \bar{Ω}$ is a W^1,p/(p−1)-extension domain.