Skip to main content
×
×
Home

A Note About the Strong Maximum Principle on RCD Spaces

  • Nicola Gigli (a1) and Chiara Rigoni (a2)
Abstract

We give a direct proof of the strong maximum principle on finite dimensional RCD spaces based on the Laplacian comparison of the squared distance.

Copyright
Footnotes
Hide All

This research has been supported by the MIUR SIR-grant ‘Nonsmooth Differential Geometry’ (RBSI147UG4).

Footnotes
References
Hide All
[1] Ambrosio, L. and Gigli, N., A user’s guide to optimal transport . In: Modelling and optimisation of flows on networks, Lecture Notes in Mathematics, 2062, Fond. CIME/CIME Found. Subser., Springer, Heidelberg, 2013, pp. 1155. https://doi.org/10.1007/978-3-642-32160-3_1.
[2] Ambrosio, L., Gigli, N., and Savaré, G., Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below . Invent. Math. 195(2014), 289391. https://doi.org/10.1007/s00222-013-0456-1.
[3] Ambrosio, L., Gigli, N., and Savaré, G., Metric measure spaces with Riemannian Ricci curvature bounded from below . Duke Math. J. 163(2014), no. 7, 14051490. https://doi.org/10.1215/00127094-2681605.
[4] Ambrosio, L., Gigli, N., and Savaré, G., Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds . Ann. Probab. 43(2015), 339404. https://doi.org/10.1214/14-AOP907.
[5] Ambrosio, L., Mondino, A., and Savaré, G., Nonlinear diffusion equations and curvature conditions in metric measure spaces. 2015. arxiv:1509.07273 https://doi.org/10.1007/s12220-014-9537-7.
[6] Björn, A. and Björn, J., Nonlinear potential theory on metric spaces. EMS Tracts in Mathematics, 17, European Mathematical Society (EMS), Zürich, 2011. https://doi.org/10.4171/099.
[7] Bogachev, V., Measure theory. Vol. I, II, Springer-Verlag, Berlin, 2007. https://doi.org/10.1007/978-3-540-34514-5.
[8] Cavalletti, F., An overview of L 1 optimal transportation on metric measure spaces. Partial Differ. Equ. Meas. Theory, De Gruyter Open, Warsaw, 2017.
[9] Cavalletti, F. and Mondino, A., Optimal maps in essentially non-branching spaces . Commun. Contemp. Math. 19(2017), no. 6, 1750007. https://doi.org/10.1142/S0219199717500079.
[10] Cheeger, J., Differentiability of Lipschitz functions on metric measure spaces . Geom. Funct. Anal. 9(1999), 428517. https://doi.org/10.1007/s000390050094.
[11] Erbar, M., Kuwada, K., and Sturm, K.-T., On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces . Invent. Math. 201(2015), 9931071. https://doi.org/10.1007/s00222-014-0563-7.
[12] Galaz-Garcia, F., Kell, M., Mondino, A., and Sosa, G., On quotients of spaces with Ricci curvature bounded below. arxiv:1704.05428.
[13] Gigli, N., Optimal maps in non branching spaces with Ricci curvature bounded from below . Geom. Funct. Anal. 22(2012), 990999. https://doi.org/10.1007/s00039-012-0176-5.
[14] Gigli, N., The splitting theorem in non-smooth context. 2013. arxiv:1302.5555.
[15] Gigli, N., An overview of the proof of the splitting theorem in spaces with non-negative Ricci curvature . Anal. Geom. Metr. Spaces 2(2014), 169213. https://doi.org/10.2478/agms-2014-0006.
[16] Gigli, N., On the differential structure of metric measure spaces and applications . Mem. Amer. Math. Soc. 236(2015), no. 1113. https://doi.org/10.1090/memo/1113.
[17] Gigli, N. and Mondino, A., A PDE approach to nonlinear potential theory in metric measure spaces . J. Math. Pures Appl. (9) 100(2013), 505534. https://doi.org/10.1016/j.matpur.2013.01.011.
[18] Gigli, N., Rajala, T., and Sturm, K.-T., Optimal maps and exponentiation on finite-dimensional spaces with Ricci curvature bounded from below . J. Geom. Anal. 26(2016), 29142929. https://doi.org/10.1007/s12220-015-9654-y.
[19] Gigli, N. and Tamanini, L., Second order differentiation formula on RCD* (K, N) spaces . Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29(2018), no. 2, 377386. https://doi.org/10.4171/RLM/811.
[20] Hopf, E., Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus. Sitzungsberichte Preussiche Akad. Wiss. (1927), pp. 147–152.
[21] Hopf, E., A remark on linear elliptic differential equations of second order . Proc. Amer. Math. Soc. 3(1952), 791793. https://doi.org/10.2307/2032182.
[22] Kell, M., Transport maps, non-branching sets of geodesics and measure rigidity . Adv. Math. 320(2017), 520573. https://doi.org/10.1016/j.aim.2017.09.003.
[23] Ohta, S.-i., On the measure contraction property of metric measure spaces . Comment. Math. Helv. 82(2007), 805828. https://doi.org/10.4171/CMH/110.
[24] Rajala, T., Local Poincaré inequalities from stable curvature conditions on metric spaces . Calc. Var. Partial Differential Equations 44(2012), 477494. https://doi.org/10.1007/s00526-011-0442-7.
[25] Rajala, T. and Sturm, K.-T., Non-branching geodesics and optimal maps in strong CD (K, )-spaces . Calc. Var. Partial Differential Equations 50(2014), 831846. https://doi.org/10.1007/s00526-013-0657-x.
[26] Santambrogio, F., Optimal transport for applied mathematicians. Calculus of variations, PDEs, and modeling. In: Progress in Nonlinear Differential Equations and their Applications, 87, Birkhäuser/Springer, Cham, 2015.
[27] Sturm, K.-T., On the geometry of metric measure spaces. II . Acta Math. 196(2006), 133177. https://doi.org/10.1007/s11511-006-0003-7.
[28] Villani, C., Optimal transport. Old and new. Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009. https://doi.org/10.1007/978-3-540-71050-9.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed