<p>The Laplace equation is considered on a part of the unit disk <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$D=\left\{ \left( r;\varphi \right) :0<r<1,\, 0<\varphi <\frac{\pi }{2} \right\} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>=</mo> <mfenced close="}" open="{"> <mfenced close=")" open="("> <mi>r</mi> <mo>;</mo> <mi>φ</mi> </mfenced> <mo>:</mo> <mn>0</mn> <mo><</mo> <mi>r</mi> <mo><</mo> <mn>1</mn> <mo>,</mo> <mspace width="0.166667em" /> <mn>0</mn> <mo><</mo> <mi>φ</mi> <mo><</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mfenced> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\left( r;\varphi \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mi>r</mi> <mo>;</mo> <mi>φ</mi> </mfenced> </math></EquationSource> </InlineEquation> are polar coordinates) subject to mixed boundary conditions, namely on the boundary segment <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Gamma _{1} =\left\{ \left( 1;\varphi \right) :\, 0<\varphi <\frac{\pi }{2} \right\} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mn>1</mn> </msub> <mo>=</mo> <mfenced close="}" open="{"> <mfenced close=")" open="("> <mn>1</mn> <mo>;</mo> <mi>φ</mi> </mfenced> <mo>:</mo> <mspace width="0.166667em" /> <mn>0</mn> <mo><</mo> <mi>φ</mi> <mo><</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, a Dirichlet condition is imposed, and on the boundary segments <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Gamma _{2} \bigcup \Gamma _{3} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mn>2</mn> </msub> <mo>⋃</mo> <msub> <mi mathvariant="normal">Γ</mi> <mn>3</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, oblique derivative conditions are imposed, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Gamma _{2} =\left\{ \left( r;0\right) :\, 0<r<1\right\} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mn>2</mn> </msub> <mo>=</mo> <mfenced close="}" open="{"> <mfenced close=")" open="("> <mi>r</mi> <mo>;</mo> <mn>0</mn> </mfenced> <mo>:</mo> <mspace width="0.166667em" /> <mn>0</mn> <mo><</mo> <mi>r</mi> <mo><</mo> <mn>1</mn> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Gamma _{3} =\left\{ \left( r;\frac{\pi }{2} \right) :\, 0<r<1\right\} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mn>3</mn> </msub> <mo>=</mo> <mfenced close="}" open="{"> <mfenced close=")" open="("> <mi>r</mi> <mo>;</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mfenced> <mo>:</mo> <mspace width="0.166667em" /> <mn>0</mn> <mo><</mo> <mi>r</mi> <mo><</mo> <mn>1</mn> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. A weighted Sobolev space <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$W_{p;\rho }^{2} \left( D\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>W</mi> <mrow> <mi>p</mi> <mo>;</mo> <mi>ρ</mi> </mrow> <mn>2</mn> </msubsup> <mfenced close=")" open="("> <mi>D</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, with weight function <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>, independent of <i>r</i>, is defined. The Noetherness of this boundary value problem in <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$W_{p;\rho }^{2} \left( D\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>W</mi> <mrow> <mi>p</mi> <mo>;</mo> <mi>ρ</mi> </mrow> <mn>2</mn> </msubsup> <mfenced close=")" open="("> <mi>D</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> (in the strong sense) is established under certain Muckenhoupt-type conditions on the weight function <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\rho \left( \cdot \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mfenced close=")" open="("> <mo>·</mo> </mfenced> </mrow> </math></EquationSource> </InlineEquation> and on the parameters of the boundary conditions. The index of the problem is also determined.</p>

ON THE NOETHERNESS OF ONE MIXED BOUNDARY VALUE PROBLEM FOR LAPLACE EQUATION IN THE WEIGHTED SOBOLEV SPACE

期刊论文
发表日期：

2025年5月1日

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Telman B. Gasymov,
Sabina R. Sadigova,
Ilkin Q. Feyzullayev

摘要

The Laplace equation is considered on a part of the unit disk $D=\left\{ \left( r;\varphi \right) :0<r<1,\, 0<\varphi <\frac{\pi }{2} \right\} $ $D = \{(r; φ) : 0 < r < 1, 0 < φ < \frac{π}{2}\}$ ( $\left( r;\varphi \right) $ $(r; φ)$ are polar coordinates) subject to mixed boundary conditions, namely on the boundary segment $\Gamma _{1} =\left\{ \left( 1;\varphi \right) :\, 0<\varphi <\frac{\pi }{2} \right\} $ $Γ_{1} = \{(1; φ) : 0 < φ < \frac{π}{2}\}$ , a Dirichlet condition is imposed, and on the boundary segments $\Gamma _{2} \bigcup \Gamma _{3} $ $Γ_{2} ⋃ Γ_{3}$ , oblique derivative conditions are imposed, where $\Gamma _{2} =\left\{ \left( r;0\right) :\, 0<r<1\right\} $ $Γ_{2} = \{(r; 0) : 0 < r < 1\}$ , $\Gamma _{3} =\left\{ \left( r;\frac{\pi }{2} \right) :\, 0<r<1\right\} $ $Γ_{3} = \{(r; \frac{π}{2}) : 0 < r < 1\}$ . A weighted Sobolev space $W_{p;\rho }^{2} \left( D\right) $ $W_{p; ρ}^{2} (D)$ , with weight function $\rho $ $ρ$ , independent of r, is defined. The Noetherness of this boundary value problem in $W_{p;\rho }^{2} \left( D\right) $ $W_{p; ρ}^{2} (D)$ (in the strong sense) is established under certain Muckenhoupt-type conditions on the weight function $\rho \left( \cdot \right) $ $ρ (\cdot)$ and on the parameters of the boundary conditions. The index of the problem is also determined.