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Cauchy problems for heat equations in metric measure spaces with non-negative Ricci curvature
- Acta Scientiarum Naturalium Universitatis Sunyatseni Vol. 60, Issue 5, Pages: 152-165(2021)
- 作者机构：
天津大学应用数学中心，天津 300072
- 作者简介：
- 基金信息：
DOI：10.13471/j.cnki.acta.snus.2020.04.30.2020B047
CLC： O175.8Published：25 September 2021，
Published Online：17 September 2020，
Received：30 April 2020，
Accepted：05 June 2020
扫 描 看 全 文
孙萌.热方程在非负Ricci曲率度量测度空间上的Cauchy问题[J].中山大学学报(自然科学版),2021,60(05):152-165.
SUN Meng.Cauchy problems for heat equations in metric measure spaces with non-negative Ricci curvature[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2021,60(05):152-165.
孙萌.热方程在非负Ricci曲率度量测度空间上的Cauchy问题[J].中山大学学报(自然科学版),2021,60(05):152-165. DOI： 10.13471/j.cnki.acta.snus.2020.04.30.2020B047.
SUN Meng.Cauchy problems for heat equations in metric measure spaces with non-negative Ricci curvature[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2021,60(05):152-165. DOI： 10.13471/j.cnki.acta.snus.2020.04.30.2020B047.
摘要
设
<math id="M1"><mo stretchy="false">(</mo><mi>X</mi><mo>
</mo><mi>d</mi><mo>
</mo><mi>μ</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49469726&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49469719&type=
10.83733368
2.96333337
是满足非负Ricci曲率条件的度量测度空间。在上半空间
<math id="M2"><mi>X</mi><mo>×</mo><msub><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mo>+</mo></mrow></msub></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49469765&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49469746&type=
9.39799976
3.21733332
上，考虑热方程的Cauchy问题。热方程为
<math id="M3"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow></msub><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo>
</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>x</mi></mrow></msub><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo>
</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn><mo>
</mo><mtext> </mtext><mi>x</mi><mo>∈</mo><mi>X</mi><mo>
</mo><mtext> </mtext><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>
</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo>
</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>
</mo><mtext> </mtext><mi>x</mi><mo>∈</mo><mi>X</mi><mo>
</mo></mtd></mtr></mtable></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49469780&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49469775&type=
67.98733521
9.90600014
其中
<math id="M4"><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>x</mi></mrow></msub></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49469807&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49469784&type=
3.13266683
3.30200005
是
<math id="M5"><mi>X</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49469817&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49469823&type=
1.94733346
2.28600001
上的Laplace算子。我们得到了：若
<math id="M6"><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo>
</mo><mi>t</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49469863&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49469849&type=
8.29733276
2.96333337
是热方程的解（称其为热函数）且满足Carleson测度条件
则它的迹
<math id="M7"><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo>
</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49469876&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49469857&type=
18.28800011
2.96333337
是有界平均振动（BMO）函数。反之，迹满足BMO条件的所有热函数
<math id="M8"><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo>
</mo><mi>t</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49469891&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49469897&type=
8.29733276
2.96333337
恰好满足Carleson测度条件
<math id="M9"><mo stretchy="false">(</mo><mi mathvariant="normal">*</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49469927&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49469905&type=
3.55599999
2.96333337
式。
（*）
<math id="M10"><munder><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">p</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>
</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>B</mi></mrow></msub></mrow></munder><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>μ</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>
</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>B</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><msubsup><mrow><mi>r</mi></mrow><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></msubsup><mrow><msub><mo>∫</mo><mrow><mi>B</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>
</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>B</mi></mrow></msub><mo stretchy="false">)</mo></mrow></msub><mrow><msup><mrow><mfenced separators="|"><mrow><mo stretchy="false">|</mo><mi>t</mi><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo stretchy="false">|</mo><mo>+</mo><mo stretchy="false">|</mo><mroot><mrow><mi>t</mi></mrow><mrow/></mroot><msub><mrow><mo>∇</mo></mrow><mrow><mi>x</mi></mrow></msub><mi>u</mi><mo stretchy="false">|</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mi mathvariant="normal">d</mi><mi>μ</mi><mfrac><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow><mrow><mi>t</mi></mrow></mfrac></mrow><mo>≤</mo><mi>C</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49469941&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49469932&type=
82.88867188
8.80533314
，
Abstract
Let
<math id="M11"><mo stretchy="false">(</mo><mi>X</mi><mo>
</mo><mi>d</mi><mo>
</mo><mi>μ</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49472936&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49472947&type=
12.61533356
3.47133350
be a metric measure space with non-negative Ricci curvature. This paper is focuses on the Cauchy problem for the heat equation on the upper half-space
<math id="M12"><mi>X</mi><mo>×</mo><msub><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mo>+</mo></mrow></msub></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49469999&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49469991&type=
10.83733368
3.80999994
.The heat equation is
<math id="M13"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow></msub><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo>
</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>x</mi></mrow></msub><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo>
</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn><mo>
</mo><mtext> </mtext><mi>x</mi><mo>∈</mo><mi>X</mi><mo>
</mo><mtext> </mtext><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo>
</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>
</mo><mtext> </mtext><mi>x</mi><mo>∈</mo><mi>X</mi><mo>
</mo></mtd></mtr></mtable></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49470033&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49470011&type=
64.34667206
9.90600014
where
<math id="M14"><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>x</mi></mrow></msub></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49470060&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49470047&type=
3.55599999
3.80999994
is the Laplace operator on
<math id="M15"><mi>X</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49471777&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49471774&type=
2.28600001
2.62466669
. We derive that a function
<math id="M16"><mi>f</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49473814&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49473801&type=
1.10066664
3.47133350
of bounded mean oscillation BMO is the trace of solution
<math id="M17"><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo>
</mo><mi>t</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49470436&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49470417&type=
9.65200043
3.47133350
of heat equation above （called Caloric function），
<math id="M18"><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo>
</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49470421&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49470410&type=
21.33600044
3.47133350
， whenever
<math id="M19"><mi>u</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49472750&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49472749&type=
1.86266661
2.62466669
satisfies the following Carleson measure condition
（*）
<math id="M20"><munder><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">p</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>
</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>B</mi></mrow></msub></mrow></munder><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>μ</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>
</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>B</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><msubsup><mrow><mi>r</mi></mrow><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></msubsup><mrow><msub><mo>∫</mo><mrow><mi>B</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>
</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>B</mi></mrow></msub><mo stretchy="false">)</mo></mrow></msub><mrow><msup><mrow><mfenced separators="|"><mrow><mo stretchy="false">|</mo><mi>t</mi><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo stretchy="false">|</mo><mo>+</mo><mo stretchy="false">|</mo><mroot><mrow><mi>t</mi></mrow><mrow/></mroot><msub><mrow><mo>∇</mo></mrow><mrow><mi>x</mi></mrow></msub><mi>u</mi><mo stretchy="false">|</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mi mathvariant="normal">d</mi><mi>μ</mi><mfrac><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow><mrow><mi>t</mi></mrow></mfrac></mrow><mo>≤</mo><mi>C</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49470163&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49470144&type=
82.88867188
8.80533314
，
Conversely， the condition
<math id="M21"><mo stretchy="false">(</mo><mi mathvariant="normal">*</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49470172&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49470168&type=
4.14866638
3.47133350
characterizes all the Caloric functions whose traces are in BMO space.
关键词
热函数度量测度空间BMOCarleson测度
Keywords
Caloric functionmetric measure spaceBMOCarleson measure
references
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FABES E B， JOHNSON R L， NERI U. Spaces of harmonic functions representable by Poisson integrals of functions in BMO and <math id="M438"><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>λ</mi></mrow></msub></math>https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49473885&type=https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49473884&type=4.656666763.21733332［J］. Indiana University Mathematics Journal， 1976， 25（2）： 159-170.
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ZHANG H C， ZHU X P. Local Li-Yau's estimates on <math id="M439"><mi mathvariant="normal">R</mi><mi mathvariant="normal">C</mi><msup><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>N</mi><mo stretchy="false">)</mo></math>https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49473886&type=https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49473874&type=15.917332653.21733332 metric measure spaces［J］. Calculus of Variations and Partial Differential Equations， 2016， 55： 30 .
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