<p>In this paper a Dirichlet boundary value problem for monogenic functions in Clifford analysis is investigated: Half of the components of the desired monogenic function can be prescribed on the whole boundary of domain <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1679_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1679_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^{n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>, while the rest components can be prescribed only on <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1679_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$(n-1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-dimensional part of the boundary. Moreover, half of these last components can be prescribed only on <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1679_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$(n-2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-dimensional part of the boundary, and so on. Finally, one component is prescribed at one point inside the domain. We prove that if the boundary datas are Hölder continuously differentiable functions, then the unique solution is Hölder continuous.</p>

Dirichlet Boundary Value Problems for Monogenic Functions of Prescribed Hölder Continuously Differentiable Data on Distinguishing Boundaries in Clifford Analysis

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发表日期：

2025年3月17日

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Dao Viet Cuong,
Wolfgang Tutschke,
Le Hung Son

摘要

In this paper a Dirichlet boundary value problem for monogenic functions in Clifford analysis is investigated: Half of the components of the desired monogenic function can be prescribed on the whole boundary of domain $\Omega $ $Ω$ in $\mathbb {R}^{n+1}$ $R^{n + 1}$ , while the rest components can be prescribed only on $(n-1)$ $(n - 1)$ -dimensional part of the boundary. Moreover, half of these last components can be prescribed only on $(n-2)$ $(n - 2)$ -dimensional part of the boundary, and so on. Finally, one component is prescribed at one point inside the domain. We prove that if the boundary datas are Hölder continuously differentiable functions, then the unique solution is Hölder continuous.