<p>In this article, we present a simpler and alternative proof of the solvability of the regularity problem—that is, the Dirichlet problem with boundary data in <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3837_Article_IEq6.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">$\dot{W}^{1,p}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mover accent="true"> <mi>W</mi> <mo>˙</mo> </mover> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>—for uniformly elliptic operators on <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3837_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {R}}^n_+$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>+</mo> <mi>n</mi> </msubsup> </math></EquationSource> </InlineEquation> under a (possibly large) Carleson measure condition. In addition, we slightly expand the class of operators for which the regularity problem is solvable, and establish an analogous result for weighted uniformly elliptic operators on <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3837_Article_IEq8.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {R}}^n \setminus {\mathbb {R}}^d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3837_Article_IEq9.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$d < n - 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo><</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

An alternative proof of the solvability of the $L^p$-regularity problem for Dahlberg-Kenig-Pipher operators on ${\mathbb {R}}^n_+$

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2025年8月29日

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Joseph Feneuil

摘要

In this article, we present a simpler and alternative proof of the solvability of the regularity problem—that is, the Dirichlet problem with boundary data in $\dot{W}^{1,p}$ ${\dot{W}}^{1, p}$ —for uniformly elliptic operators on ${\mathbb {R}}^n_+$ $R_{+}^{n}$ under a (possibly large) Carleson measure condition. In addition, we slightly expand the class of operators for which the regularity problem is solvable, and establish an analogous result for weighted uniformly elliptic operators on ${\mathbb {R}}^n \setminus {\mathbb {R}}^d$ $R^{n} \ R^{d}$ , where $d < n - 1$ $d < n - 1$ .

An alternative proof of the solvability of the \(L^p\)-regularity problem for Dahlberg-Kenig-Pipher operators on \({\mathbb {R}}^n_+\)

摘要