热方程在非负Ricci曲率度量测度空间上的Cauchy问题

孙萌

doi:10.13471/j.cnki.acta.snus.2020.04.30.2020B047

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热方程在非负Ricci曲率度量测度空间上的Cauchy问题

研究论文 | 更新时间：2023-11-01

- 热方程在非负Ricci曲率度量测度空间上的Cauchy问题
- Cauchy problems for heat equations in metric measure spaces with non-negative Ricci curvature
- 中山大学学报(自然科学版) 2021年60卷第5期页码：152-165
- 作者机构：
  
  天津大学应用数学中心，天津 300072
- 作者简介：
  
  孙萌（1995年生），女；研究方向：调和分析；E-mail：Sunmeng_666@163.com
- 基金信息：
  
  国家自然科学基金(11922114;11671039;11771043)
- DOI：10.13471/j.cnki.acta.snus.2020.04.30.2020B047
  中图分类号： O175.8
- 纸质出版日期：2021-09-25，
  
  网络出版日期：2020-09-17，
  
  收稿日期：2020-04-30，
  
  录用日期：2020-06-05
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

</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

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

</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

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

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

</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

</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

</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

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。

（*）

</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

</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

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

</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

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

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

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

</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），

</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

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

（*）

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

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

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FABES E B， NERI U. Characterization of temperatures with initial data in BMO［J］. Duke Mathematical Journal， 1975， 42： 725-734.

DUONG X T， YAN L X， ZHANG C. On characterization of Poisson integrals of Schrödinger operators with BMO traces［J］. Journal of Functional Analysis， 2014， 266（4）： 2053-2085.

JIANG R J， XIAO J， YANG D C. Towards spaces of harmonic functions with traces in square Campanato spaces and their scaling invariants［J］. Analysis and Applications， 2016， 14（5）： 679-703.

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.

CHEN J C. A representation theorem of harmonic functions and its application to BMO on manifolds［J］. Applied Mathematics， 2001， 16： 279-284.

ZHANG H C， ZHU X P. On a new definition of Ricci curvature on Alexandrov spaces［J］. Acta Mathematica Scientia， Series B（English Edition）， 2010， 30（6）： 1949-1974.

ZHANG H C， ZHU X P. Yau's gradient estimates on Alexandrov spaces［J］. Journal of Differential Geometry， 2012， 91（3）：445-522.

AMBROSIO L， GIGLI N， SAVARÉ G. Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds［J］. The Annals of Probability， 2015， 43（1）： 339-404.

AMBROSIO L， GIGLI N， SAVARÉ G. Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces［J］. Revista Matemática Iberoamericana， 2013， 29： 969-996.

JIANG R J. The Li-Yau inequality and heat kernels on metric measure spaces［J］. Journal de Mathématiques Pures et Appliquées， 2015， 104（1）： 29-57.

GIGLI N. On the differential structure of metric measure spaces and applications［J/OL］. （2013-05-21）［2020-04-30］. https：//arxiv.org/pdf/1205.6622.pdfhttps://arxiv.org/pdf/1205.6622.pdf.

ERBAR M， KUWADA K， STURM K T. On the equivalence of the entropic curvature-dimension condition and Bochner's inequality on metric measure spaces［J］. Inventiones Mathematicae， 2015， 201： 993-1071.

HOFMANN S， LU G， MITREA D， et al. Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates ［J/OL］. Memoirs of the American Mathematical Society， 2011， 214（1007）： 1-84.

STURM K T. On the geometry of metric measure spaces II［J］. Acta Mathematica， 2006， 196： 133-177.

RAJALA T. Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm［J］. Journal of Functional Analysis， 2012， 263： 896-924.

JIANG R J， LI H Q， ZHANG H C. Heat kernel bounds on metric measure spaces and some applications［J］. Potential Analysis， 2016， 44（3）： 601-627.

DUONG X T， YAN L X. New function spaces of BMO type， the John-Nirenberg inequality， interpolation， and applications［J］. Communications on Pure and Applied Mathematics， 2005， 58（10）： 1375-1420.

STURM K T. Analysis on local dirichlet spaces III ［J］. Journal de Mathématiques Pures et Appliquées，1996， 75（3）： 273-297.

HOU Q， SALOFF-COSTE L. Time regularity for local weak solutions of the heat equation on local Dirichlet spaces［J/OL］. （2019-11-30）［2020-04-03］.https：//arxiv.org/abs/1912.12998https://arxiv.org/abs/1912.12998.

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 .

DZIUBAŃSKI J， GARRIGÓS G， MARTÍNEZ T， et al. BMO spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality［J］. Mathematische Zeitschrift， 2005， 249： 329-356.

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