Geodesic
A Note About the Strong Maximum Principle on RCD Spaces
Canadian mathematical bulletin, Tome 62 (2019) no. 2, pp. 259-266

Voir la notice de l'article provenant de la source Cambridge University Press

We give a direct proof of the strong maximum principle on finite dimensional RCD spaces based on the Laplacian comparison of the squared distance.
DOI : 10.4153/CMB-2018-022-9
Mots-clés : maximum principle, RCD space 
Gigli, Nicola; Rigoni, Chiara. A Note About the Strong Maximum Principle on RCD Spaces. Canadian mathematical bulletin, Tome 62 (2019) no. 2, pp. 259-266. doi: 10.4153/CMB-2018-022-9
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[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. 1–155. . Google Scholar | DOI

[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), 289–391. . Google Scholar | DOI

[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, 1405–1490. . Google Scholar | DOI

[4] Ambrosio, L., Gigli, N., and Savaré, G., Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds . Ann. Probab. 43(2015), 339–404. . Google Scholar | DOI

[5] Ambrosio, L., Mondino, A., and Savaré, G., Nonlinear diffusion equations and curvature conditions in metric measure spaces. 2015. arxiv:1509.07273  . Google Scholar | DOI

[6] Björn, A. and Björn, J., Nonlinear potential theory on metric spaces. EMS Tracts in Mathematics, 17, European Mathematical Society (EMS), Zürich, 2011. . Google Scholar | DOI

[7] Bogachev, V., Measure theory. Vol. I, II, Springer-Verlag, Berlin, 2007. . Google Scholar | DOI

[8] 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

[9] Cavalletti, F. and Mondino, A., Optimal maps in essentially non-branching spaces . Commun. Contemp. Math. 19(2017), no. 6, 1750007. . Google Scholar | DOI

[10] Cheeger, J., Differentiability of Lipschitz functions on metric measure spaces . Geom. Funct. Anal. 9(1999), 428–517. . Google Scholar | DOI

[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), 993–1071. . Google Scholar | DOI

[12] Galaz-Garcia, F., Kell, M., Mondino, A., and Sosa, G., On quotients of spaces with Ricci curvature bounded below. arxiv:1704.05428. Google Scholar

[13] Gigli, N., Optimal maps in non branching spaces with Ricci curvature bounded from below . Geom. Funct. Anal. 22(2012), 990–999. . Google Scholar | DOI

[14] Gigli, N., The splitting theorem in non-smooth context. 2013. arxiv:1302.5555. Google Scholar

[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), 169–213. . Google Scholar | DOI

[16] Gigli, N., On the differential structure of metric measure spaces and applications . Mem. Amer. Math. Soc. 236(2015), no. 1113. . Google Scholar | DOI

[17] 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. . Google Scholar | DOI

[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), 2914–2929. . Google Scholar | DOI

[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, 377–386. . Google Scholar | DOI

[20] Hopf, E., Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus. Sitzungsberichte Preussiche Akad. Wiss. (1927), pp. 147–152. Google Scholar

[21] Hopf, E., A remark on linear elliptic differential equations of second order . Proc. Amer. Math. Soc. 3(1952), 791–793. . Google Scholar | DOI

[22] Kell, M., Transport maps, non-branching sets of geodesics and measure rigidity . Adv. Math. 320(2017), 520–573. . Google Scholar | DOI

[23] Ohta, S.-I., On the measure contraction property of metric measure spaces . Comment. Math. Helv. 82(2007), 805–828. . Google Scholar | DOI

[24] Rajala, T., Local Poincaré inequalities from stable curvature conditions on metric spaces . Calc. Var. Partial Differential Equations 44(2012), 477–494. . Google Scholar | DOI

[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), 831–846. . Google Scholar | DOI

[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. Google Scholar

[27] Sturm, K.-T., On the geometry of metric measure spaces. II . Acta Math. 196(2006), 133–177. . Google Scholar | DOI

[28] Villani, C., Optimal transport. Old and new. Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009. . Google Scholar | DOI

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