Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-24T19:39:37.485Z Has data issue: false hasContentIssue false
  • English
  • Français

A Note About the Strong Maximum Principle on RCD Spaces

Part of: Partial differential equations Potential theory on metric spaces

Published online by Cambridge University Press:  07 January 2019

Nicola Gigli  and
Chiara Rigoni
Nicola Gigli
Affiliation:
SISSA, 34136 Trieste, Italy Email: ngigli@sissa.it
Chiara Rigoni
Affiliation:
Hausdorff center for Mathematics, D-53115 Bonn, Germany Email: crigoni@sissa.it
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

Keywords

maximum principleRCD space

MSC classification

Primary: 31E05: Potential theory on metric spaces
Secondary: 35B50: Maximum principles
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 2 , June 2019 , pp. 259 - 266
Copyright
© Canadian Mathematical Society 2018 

Footnotes

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

References

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. 1–155. https://doi.org/10.1007/978-3-642-32160-3_1.Google Scholar
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), 289–391. https://doi.org/10.1007/s00222-013-0456-1.Google Scholar
Ambrosio, L., Gigli, N., and Savaré, G., Metric measure spaces with Riemannian Ricci curvature bounded from below . Duke Math. J. 163(2014), no. 7, 1405–1490. https://doi.org/10.1215/00127094-2681605.Google Scholar
Ambrosio, L., Gigli, N., and Savaré, G., Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds . Ann. Probab. 43(2015), 339–404. https://doi.org/10.1214/14-AOP907.Google Scholar
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.Google Scholar
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.Google Scholar
Bogachev, V., Measure theory. Vol. I, II, Springer-Verlag, Berlin, 2007. https://doi.org/10.1007/978-3-540-34514-5.Google Scholar
Cavalletti, F., An overview of L 1 optimal transportation on metric measure spaces. Partial Differ. Equ. Meas. Theory, De Gruyter Open, Warsaw, 2017.Google Scholar
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.Google Scholar
Cheeger, J., Differentiability of Lipschitz functions on metric measure spaces . Geom. Funct. Anal. 9(1999), 428–517. https://doi.org/10.1007/s000390050094.Google Scholar
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), 993–1071. https://doi.org/10.1007/s00222-014-0563-7.Google Scholar
Galaz-Garcia, F., Kell, M., Mondino, A., and Sosa, G., On quotients of spaces with Ricci curvature bounded below. arxiv:1704.05428.Google Scholar
Gigli, N., Optimal maps in non branching spaces with Ricci curvature bounded from below . Geom. Funct. Anal. 22(2012), 990–999. https://doi.org/10.1007/s00039-012-0176-5.Google Scholar
Gigli, N., The splitting theorem in non-smooth context. 2013. arxiv:1302.5555.Google Scholar
Gigli, N., An overview of the proof of the splitting theorem in spaces with non-negative Ricci curvature . Anal. Geom. Metr. Spaces 2(2014), 169–213. https://doi.org/10.2478/agms-2014-0006.Google Scholar
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.Google Scholar
Gigli, N. and Mondino, A., A PDE approach to nonlinear potential theory in metric measure spaces . J. Math. Pures Appl. (9) 100(2013), 505–534. https://doi.org/10.1016/j.matpur.2013.01.011.Google Scholar
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), 2914–2929. https://doi.org/10.1007/s12220-015-9654-y.Google Scholar
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, 377–386. https://doi.org/10.4171/RLM/811.Google Scholar
Hopf, E., Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus. Sitzungsberichte Preussiche Akad. Wiss. (1927), pp. 147–152.Google Scholar
Hopf, E., A remark on linear elliptic differential equations of second order . Proc. Amer. Math. Soc. 3(1952), 791–793. https://doi.org/10.2307/2032182.Google Scholar
Kell, M., Transport maps, non-branching sets of geodesics and measure rigidity . Adv. Math. 320(2017), 520–573. https://doi.org/10.1016/j.aim.2017.09.003.Google Scholar
Ohta, S.-i., On the measure contraction property of metric measure spaces . Comment. Math. Helv. 82(2007), 805–828. https://doi.org/10.4171/CMH/110.Google Scholar
Rajala, T., Local Poincaré inequalities from stable curvature conditions on metric spaces . Calc. Var. Partial Differential Equations 44(2012), 477–494. https://doi.org/10.1007/s00526-011-0442-7.Google Scholar
Rajala, T. and Sturm, K.-T., Non-branching geodesics and optimal maps in strong CD (K, ∞)-spaces . Calc. Var. Partial Differential Equations 50(2014), 831–846. https://doi.org/10.1007/s00526-013-0657-x.Google Scholar
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.Google Scholar
Sturm, K.-T., On the geometry of metric measure spaces. II . Acta Math. 196(2006), 133–177. https://doi.org/10.1007/s11511-006-0003-7.Google Scholar
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.Google Scholar