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A Runge theorem for solutions of the heat equation
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| Zeitschriftentitel: | Proceedings of the American Mathematical Society |
|---|---|
| Personen und Körperschaften: | |
| In: | Proceedings of the American Mathematical Society, 80, 1980, 4, S. 643-646 |
| Format: | E-Article |
| Sprache: | Englisch |
| veröffentlicht: |
American Mathematical Society (AMS)
|
| Schlagwörter: |
| author_facet |
Diaz, R. Diaz, R. |
|---|---|
| author |
Diaz, R. |
| spellingShingle |
Diaz, R. Proceedings of the American Mathematical Society A Runge theorem for solutions of the heat equation Applied Mathematics General Mathematics |
| author_sort |
diaz, r. |
| spelling |
Diaz, R. 0002-9939 1088-6826 American Mathematical Society (AMS) Applied Mathematics General Mathematics http://dx.doi.org/10.1090/s0002-9939-1980-0587944-2 <p>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be open sets in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{R^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 1 subset-of normal upper Omega 2"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _1} \subset {\Omega _2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Every solution of the heat equation on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> admits approximation on the compact subsets of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by functions which satisfy the heat equation throughout <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if this topological condition is met: For every hyperplane <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi>π<!-- π --></mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{R^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> orthogonal to the time axis, every compact component of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi minus normal upper Omega 1"> <mml:semantics> <mml:mrow> <mml:mi>π<!-- π --></mml:mi> <mml:mi class="MJX-variant" mathvariant="normal">∖<!-- ∖ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi \backslash {\Omega _1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains a compact component of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi minus normal upper Omega 2"> <mml:semantics> <mml:mrow> <mml:mi>π<!-- π --></mml:mi> <mml:mi class="MJX-variant" mathvariant="normal">∖<!-- ∖ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi \backslash {\Omega _2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p> A Runge theorem for solutions of the heat equation Proceedings of the American Mathematical Society |
| doi_str_mv |
10.1090/s0002-9939-1980-0587944-2 |
| facet_avail |
Online Free |
| finc_class_facet |
Mathematik |
| format |
ElectronicArticle |
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| id |
ai-49-aHR0cDovL2R4LmRvaS5vcmcvMTAuMTA5MC9zMDAwMi05OTM5LTE5ODAtMDU4Nzk0NC0y |
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DE-D275 DE-Bn3 DE-Brt1 DE-Zwi2 DE-D161 DE-Gla1 DE-Zi4 DE-15 DE-Pl11 DE-Rs1 DE-105 DE-14 DE-Ch1 DE-L229 |
| imprint |
American Mathematical Society (AMS), 1980 |
| imprint_str_mv |
American Mathematical Society (AMS), 1980 |
| issn |
0002-9939 1088-6826 |
| issn_str_mv |
0002-9939 1088-6826 |
| language |
English |
| mega_collection |
American Mathematical Society (AMS) (CrossRef) |
| match_str |
diaz1980arungetheoremforsolutionsoftheheatequation |
| publishDateSort |
1980 |
| publisher |
American Mathematical Society (AMS) |
| recordtype |
ai |
| record_format |
ai |
| series |
Proceedings of the American Mathematical Society |
| source_id |
49 |
| title |
A Runge theorem for solutions of the heat equation |
| title_unstemmed |
A Runge theorem for solutions of the heat equation |
| title_full |
A Runge theorem for solutions of the heat equation |
| title_fullStr |
A Runge theorem for solutions of the heat equation |
| title_full_unstemmed |
A Runge theorem for solutions of the heat equation |
| title_short |
A Runge theorem for solutions of the heat equation |
| title_sort |
a runge theorem for solutions of the heat equation |
| topic |
Applied Mathematics General Mathematics |
| url |
http://dx.doi.org/10.1090/s0002-9939-1980-0587944-2 |
| publishDate |
1980 |
| physical |
643-646 |
| description |
<p>Let <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 1">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{\Omega _1}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> and <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 2">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{\Omega _2}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> be open sets in <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript n">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msup>
<mml:mi>R</mml:mi>
<mml:mi>n</mml:mi>
</mml:msup>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{R^n}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> such that <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 1 subset-of normal upper Omega 2">
<mml:semantics>
<mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:mrow>
<mml:mo>⊂<!-- ⊂ --></mml:mo>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:mrow>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{\Omega _1} \subset {\Omega _2}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. Every solution of the heat equation on <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 1">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{\Omega _1}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> admits approximation on the compact subsets of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 1">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{\Omega _1}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> by functions which satisfy the heat equation throughout <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 2">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{\Omega _2}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> if and only if this topological condition is met: For every hyperplane <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi">
<mml:semantics>
<mml:mi>π<!-- π --></mml:mi>
<mml:annotation encoding="application/x-tex">\pi</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> in <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript n">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msup>
<mml:mi>R</mml:mi>
<mml:mi>n</mml:mi>
</mml:msup>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{R^n}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> orthogonal to the time axis, every compact component of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi minus normal upper Omega 1">
<mml:semantics>
<mml:mrow>
<mml:mi>π<!-- π --></mml:mi>
<mml:mi class="MJX-variant" mathvariant="normal">∖<!-- ∖ --></mml:mi>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:mrow>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\pi \backslash {\Omega _1}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> contains a compact component of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi minus normal upper Omega 2">
<mml:semantics>
<mml:mrow>
<mml:mi>π<!-- π --></mml:mi>
<mml:mi class="MJX-variant" mathvariant="normal">∖<!-- ∖ --></mml:mi>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:mrow>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\pi \backslash {\Omega _2}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>.</p> |
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1792329276740075529 |
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not assigned |
| last_indexed |
2024-03-01T13:06:31.495Z |
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not assigned |
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url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fvufind.svn.sourceforge.net%3Agenerator&rft.title=A+Runge+theorem+for+solutions+of+the+heat+equation&rft.date=1980-01-01&genre=article&issn=1088-6826&volume=80&issue=4&spage=643&epage=646&pages=643-646&jtitle=Proceedings+of+the+American+Mathematical+Society&atitle=A+Runge+theorem+for+solutions+of+the+heat+equation&aulast=Diaz&aufirst=R.&rft_id=info%3Adoi%2F10.1090%2Fs0002-9939-1980-0587944-2&rft.language%5B0%5D=eng |
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| author | Diaz, R. |
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| container_volume | 80 |
| description | <p>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be open sets in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{R^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 1 subset-of normal upper Omega 2"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _1} \subset {\Omega _2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Every solution of the heat equation on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> admits approximation on the compact subsets of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by functions which satisfy the heat equation throughout <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if this topological condition is met: For every hyperplane <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi>π<!-- π --></mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{R^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> orthogonal to the time axis, every compact component of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi minus normal upper Omega 1"> <mml:semantics> <mml:mrow> <mml:mi>π<!-- π --></mml:mi> <mml:mi class="MJX-variant" mathvariant="normal">∖<!-- ∖ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi \backslash {\Omega _1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains a compact component of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi minus normal upper Omega 2"> <mml:semantics> <mml:mrow> <mml:mi>π<!-- π --></mml:mi> <mml:mi class="MJX-variant" mathvariant="normal">∖<!-- ∖ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi \backslash {\Omega _2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p> |
| doi_str_mv | 10.1090/s0002-9939-1980-0587944-2 |
| facet_avail | Online, Free |
| finc_class_facet | Mathematik |
| format | ElectronicArticle |
| format_de105 | Article, E-Article |
| format_de14 | Article, E-Article |
| format_de15 | Article, E-Article |
| format_de520 | Article, E-Article |
| format_de540 | Article, E-Article |
| format_dech1 | Article, E-Article |
| format_ded117 | Article, E-Article |
| format_degla1 | E-Article |
| format_del152 | Buch |
| format_del189 | Article, E-Article |
| format_dezi4 | Article |
| format_dezwi2 | Article, E-Article |
| format_finc | Article, E-Article |
| format_nrw | Article, E-Article |
| geogr_code | not assigned |
| geogr_code_person | not assigned |
| id | ai-49-aHR0cDovL2R4LmRvaS5vcmcvMTAuMTA5MC9zMDAwMi05OTM5LTE5ODAtMDU4Nzk0NC0y |
| imprint | American Mathematical Society (AMS), 1980 |
| imprint_str_mv | American Mathematical Society (AMS), 1980 |
| institution | DE-D275, DE-Bn3, DE-Brt1, DE-Zwi2, DE-D161, DE-Gla1, DE-Zi4, DE-15, DE-Pl11, DE-Rs1, DE-105, DE-14, DE-Ch1, DE-L229 |
| issn | 0002-9939, 1088-6826 |
| issn_str_mv | 0002-9939, 1088-6826 |
| language | English |
| last_indexed | 2024-03-01T13:06:31.495Z |
| match_str | diaz1980arungetheoremforsolutionsoftheheatequation |
| mega_collection | American Mathematical Society (AMS) (CrossRef) |
| physical | 643-646 |
| publishDate | 1980 |
| publishDateSort | 1980 |
| publisher | American Mathematical Society (AMS) |
| record_format | ai |
| recordtype | ai |
| series | Proceedings of the American Mathematical Society |
| source_id | 49 |
| spelling | Diaz, R. 0002-9939 1088-6826 American Mathematical Society (AMS) Applied Mathematics General Mathematics http://dx.doi.org/10.1090/s0002-9939-1980-0587944-2 <p>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be open sets in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{R^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 1 subset-of normal upper Omega 2"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _1} \subset {\Omega _2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Every solution of the heat equation on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> admits approximation on the compact subsets of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by functions which satisfy the heat equation throughout <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if this topological condition is met: For every hyperplane <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi>π<!-- π --></mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{R^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> orthogonal to the time axis, every compact component of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi minus normal upper Omega 1"> <mml:semantics> <mml:mrow> <mml:mi>π<!-- π --></mml:mi> <mml:mi class="MJX-variant" mathvariant="normal">∖<!-- ∖ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi \backslash {\Omega _1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains a compact component of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi minus normal upper Omega 2"> <mml:semantics> <mml:mrow> <mml:mi>π<!-- π --></mml:mi> <mml:mi class="MJX-variant" mathvariant="normal">∖<!-- ∖ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi \backslash {\Omega _2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p> A Runge theorem for solutions of the heat equation Proceedings of the American Mathematical Society |
| spellingShingle | Diaz, R., Proceedings of the American Mathematical Society, A Runge theorem for solutions of the heat equation, Applied Mathematics, General Mathematics |
| title | A Runge theorem for solutions of the heat equation |
| title_full | A Runge theorem for solutions of the heat equation |
| title_fullStr | A Runge theorem for solutions of the heat equation |
| title_full_unstemmed | A Runge theorem for solutions of the heat equation |
| title_short | A Runge theorem for solutions of the heat equation |
| title_sort | a runge theorem for solutions of the heat equation |
| title_unstemmed | A Runge theorem for solutions of the heat equation |
| topic | Applied Mathematics, General Mathematics |
| url | http://dx.doi.org/10.1090/s0002-9939-1980-0587944-2 |