Kolesnikov, Alexander V. Weak regularity of Gauss mass transport. (English) Zbl 1286.35059 Bull. Sci. Math. 138, No. 2, 165-198 (2014). Summary: Given two probability measures \(\mu\) and \(\nu\) we consider a mass transportation mapping \(T\) satisfying 1) \(T\) sends \(\mu\) to \(\nu\), 2) \(T\) has the form \(T=\varphi\frac{\nabla\varphi}{|\nabla\varphi|}\), where \(\varphi\) is a function with convex sublevel sets. We prove a change of variables formula for \(T\). We also establish Sobolev estimates for \(\varphi\), and a new form of the parabolic maximum principle. In addition, we discuss relations to the Monge-Kantorovich problem, curvature flows theory, and parabolic nonlinear PDEs. MSC: 35B65 Smoothness and regularity of solutions to PDEs 35K96 Parabolic Monge-Ampère equations 35B50 Maximum principles in context of PDEs 35B45 A priori estimates in context of PDEs 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) 35K93 Quasilinear parabolic equations with mean curvature operator Keywords:optimal transportation; Monge-Kantorovich problem; Gauss curvature flows; Alexandrov maximum principle; parabolic maximum principle; Sobolev and Hölder a priori estimates PDFBibTeX XMLCite \textit{A. V. Kolesnikov}, Bull. Sci. Math. 138, No. 2, 165--198 (2014; Zbl 1286.35059) Full Text: DOI arXiv References: [1] Ambrosio, L.; Fusco, N.; Pallara, D., Functions of Bounded Variations and Free Discontinuity Problems, Oxford Math. Monogr. (2000), Clarendon Press: Clarendon Press Oxford [2] Ambrosio, L.; Gigli, N.; Savaré, G., Gradient Flows in Metric Spaces and in the Wasserstein Spaces of Probability Measures, Lectures Math. ETH Zurich (2005), Birkhäuser Verlag: Birkhäuser Verlag Basel [3] Andrews, B., Motion of hypersurfaces by Gauss curvature, Pacific J. Math., 195, 1, 1-34 (2000) · Zbl 1028.53072 [4] Barthe, F.; Kolesnikov, A. V., Mass transport and variants of the logarithmic Sobolev inequality, J. Geom. Anal., 18, 4, 921-979 (2008) · Zbl 1170.46031 [5] Bogachev, V. I., Measure Theory, vols. 1, 2 (2007), Springer-Verlag: Springer-Verlag Berlin-New York [6] Bogachev, V. I., Differentiable Measures and the Malliavin Calculus (2010), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1247.28001 [7] Bogachev, V. I.; Kolesnikov, A. V., On the Monge-Ampère equation in infinite dimensions, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 8, 4, 547-572 (2005) · Zbl 1103.49001 [8] Bogachev, V. I.; Kolesnikov, A. V., Transformations of measures by Gauss maps, Dokl. RAN, 422, 4, 446-449 (2008) · Zbl 1267.28004 [9] Bogachev, V. I.; Kolesnikov, A. V., Mass transport generated by a flow of Gauss maps, J. Funct. Anal., 256, 3, 940-957 (2009) · Zbl 1154.49013 [10] Bogachev, V. I.; Kolesnikov, A. V.; Medvedev, K. V., Triangular transformations of measures, Sb. Math., 196, 3, 3-30 (2005) · Zbl 1072.28010 [11] Brakke, K., The Motion of a Surface by Its Mean Curvature (1978), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0386.53047 [12] Caffarelli, L. A.; Cabré, X., Fully Nonlinear Elliptic Equations (1995), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0834.35002 [13] Caffarelli, L. A., A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity, Ann. of Math. (2), 131, 1, 129-134 (1990) · Zbl 0704.35045 [14] Evans, L.; Gariepy, R. F., Measure Theory and Fine Properties of Functions (1992), CRC Press: CRC Press Boca Raton · Zbl 0804.28001 [15] Firey, W. J., Shapes of worn stones, Mathematika, 21, 1-11 (1974) · Zbl 0311.52003 [16] Gerhard, C., Curvature Problems, Ser. Geom. Topol., vol. 39 (2006), International Press [17] Giga, Y., Surface Evolution Equations. A Level Sets Approach (2006), Birkhäuser Verlag: Birkhäuser Verlag Basel [18] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (2001), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 1042.35002 [19] Gutiérrez, C. E., The Monge-Ampère Equation, Progr. Nonlinear Differential Equations Appl., vol. 44 (2001), Birkhäuser · Zbl 0989.35052 [20] Gutiérrez, C. E.; Huang, Q., \(W^{2, p}\) estimates for the parabolic Monge-Ampère equation, Arch. Ration. Mech. Anal., 159, 137-177 (2001) · Zbl 0992.35020 [21] Huisken, G., Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., 20, 237-266 (1984) · Zbl 0556.53001 [22] Ilmanen, T., Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature, J. Differential Geom., 38, 417-461 (1993) · Zbl 0784.53035 [23] Ivochkina, N. M.; Ladyzhenskaya, O. A., Parabolic equations generated by symmetric functions of the eigenvalues of the Hessian or by the principal curvatures of a surface. I. Parabolic Monge-Ampère equations, Algebra i Analiz, 6, 3, 141-160 (1994) · Zbl 0820.35083 [24] Knothe, H., Contributions to the theory of convex bodies, Michigan Math. J., 4, 39-52 (1957) · Zbl 0077.35803 [25] Krylov, N. V., Nonlinear Elliptic and Parabolic Equations of the Second Order (1987), Reidel: Reidel Dordrecht · Zbl 0619.35004 [26] Krylov, N. V., Fully nonlinear second order elliptic equations: recent developments, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), XXV, 569-595 (1997) · Zbl 1033.35036 [27] Krylov, N. V., Lectures on Elliptic and Parabolic Estimates in Hölder Spaces (1998), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI [28] Li, Chen; Lian, S. Zh.; Wang, G. L., Convex monotone functions and generalized solutions of parabolic Monge-Ampère equations, J. Differential Equations, 186, 558-571 (2002) · Zbl 1014.35042 [29] McCann, R. J., A convexity principle for interacting gases, Adv. Math., 128, 153-179 (1997) · Zbl 0901.49012 [30] Schulze, F., Nonlinear evolution by mean curvature and isoperimetric inequalities, J. Differential Geom., 79, 197-241 (2008) · Zbl 1202.53066 [31] Soner, H. M., Ginzburg-Landau equation and motion by mean curvature. II: development of the initial interface, J. Geom. Anal., 7, 476-491 (1997) · Zbl 0935.35061 [32] Spiliotis, J., Certain results on a parabolic type Monge-Ampère equation, J. Math. Anal. Appl., 163, 2, 484-511 (1992) · Zbl 0753.35044 [33] Topping, P., Mean curvature flow and geometric inequalities, J. Reine Angew. Math., 503, 47-61 (1998) · Zbl 0909.53044 [34] Tso, K., Deforming a hypersurfaces by Gauss-Kronecker curvature, Comm. Pure Appl. Math., 38, 876-882 (1985) · Zbl 0612.53005 [35] Tso, K., On the Alexandrov-Bakelman type maximum principle for second order parabolic equations, Comm. Partial Differential Equations, 10, 543-553 (1985) · Zbl 0581.35027 [36] Villani, C., Topics in Optimal Transportation (2003), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1106.90001 [37] Villani, C., Optimal Transport: Old and New (2009), Springer-Verlag: Springer-Verlag Berlin · Zbl 1156.53003 [38] Wang, R. H.; Wang, G. L., On existence, uniqueness and regularity of viscosity solutions for the first initial boundary value problems to parabolic Monge-Ampère equation, Northeast. Math. J., 8, 4, 417-446 (1992) · Zbl 0783.35028 [39] Wang, R. H.; Wang, G. L., The geometric measure theoretical characterization of viscosity solutions to parabolic Monge-Ampère type equation, J. Partial Differential Equations, 6, 3, 237-254 (1993) · Zbl 0811.35053 [40] Wang, Lihe, On the regularity theory of fully nonlinear parabolic equations. I-III, Comm. Pure Appl. Math., 45, 27-76 (1992), 141-178, 255-262 · Zbl 0832.35025 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.