<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\{v_{\alpha }\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>v</mi> <mi>α</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> be a system of polynomial solutions of the parabolic equation <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$a_{hk}\partial _{x_{h}x_{k}}u - \partial _t u =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mrow> <mi mathvariant="italic">hk</mi> </mrow> </msub> <msub> <mi>∂</mi> <mrow> <msub> <mi>x</mi> <mi>h</mi> </msub> <msub> <mi>x</mi> <mi>k</mi> </msub> </mrow> </msub> <mi>u</mi> <mo>-</mo> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>u</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> in a bounded <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-cylinder <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Omega _{T}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Ω</mi> <mi>T</mi> </msub> </math></EquationSource> </InlineEquation> contained in <InlineEquation ID="IEq5"> <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>. Here, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$a_{hk}\partial _{x_{h}x_{k}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mrow> <mi mathvariant="italic">hk</mi> </mrow> </msub> <msub> <mi>∂</mi> <mrow> <msub> <mi>x</mi> <mi>h</mi> </msub> <msub> <mi>x</mi> <mi>k</mi> </msub> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> is an elliptic operator with real constant coefficients. We prove that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\{v_{\alpha }\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>v</mi> <mi>α</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> is complete in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$L^{p}(\Sigma ')$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Σ</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Sigma '$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="normal">Σ</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> is the parabolic boundary of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\Omega _{T}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Ω</mi> <mi>T</mi> </msub> </math></EquationSource> </InlineEquation>. Similar results are proved for the adjoint equation <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$a_{hk}\partial _{x_{h}x_{k}} u+ \partial _t u =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mrow> <mi mathvariant="italic">hk</mi> </mrow> </msub> <msub> <mi>∂</mi> <mrow> <msub> <mi>x</mi> <mi>h</mi> </msub> <msub> <mi>x</mi> <mi>k</mi> </msub> </mrow> </msub> <mi>u</mi> <mo>+</mo> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>u</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

Completeness Theorems on the Boundary for a Parabolic Equation

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2026年4月27日

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A. Cialdea,
C. S. Mare

摘要

Let $\{v_{\alpha }\}$ ${v_{α}}$ be a system of polynomial solutions of the parabolic equation $a_{hk}\partial _{x_{h}x_{k}}u - \partial _t u =0$ $a_{hk} \partial_{x_{h} x_{k}} u - \partial_{t} u = 0$ in a bounded $C^1$ $C^{1}$ -cylinder $\Omega _{T}$ $Ω_{T}$ contained in $\mathbb {R}^{n+1}$ $R^{n + 1}$ . Here, $a_{hk}\partial _{x_{h}x_{k}}$ $a_{hk} \partial_{x_{h} x_{k}}$ is an elliptic operator with real constant coefficients. We prove that $\{v_{\alpha }\}$ ${v_{α}}$ is complete in $L^{p}(\Sigma ')$ $L^{p} (Σ^{'})$ , where $\Sigma '$ $Σ^{'}$ is the parabolic boundary of $\Omega _{T}$ $Ω_{T}$ . Similar results are proved for the adjoint equation $a_{hk}\partial _{x_{h}x_{k}} u+ \partial _t u =0$ $a_{hk} \partial_{x_{h} x_{k}} u + \partial_{t} u = 0$ .