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08 Nov 2012
A microscopic convexity principle for spacetime convex solutions of fully nonlinear parabolic equations
 Chuan Qiang Chen,
 Bo Wen Hu
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We study microscopic spacetime convexity properties of fully nonlinear parabolic partial differential equations. Under certain general structure condition, we establish a constant rank theorem for the spacetime convex solutions of fully nonlinear parabolic equations. At last, we consider the parabolic convexity of solutions to parabolic equations and the convexity of the spacetime second fundamental form of geometric flows.
The first author is supported by National Natural Science Foundation of China (Grant No. 10871187)
 Borell, C (1982) Brownian motion in a convex ring and quasiconcavity. Comm. Math. Phys. 86: pp. 143147 CrossRef
 Borell, C (1996) A note on parabolic convexity and heat conduction. Ann. Inst. H. Poincaré Probab. Statist. 32: pp. 387393
 Borell, C (2000) Diffusion equations and geometric inequalities. Potential Anal. 12: pp. 4971 CrossRef
 Li, P, Yau, S T (1986) On the parabolic kernel of the Schrödinger operator. Acta Math. 156: pp. 153201 CrossRef
 Hamilton, R (1993) A matrix Harnack estimate for the heat equation. Comm. Anal. Geom. 1: pp. 113126
 Hamilton, R (1995) Harnack estimate for the mean curvature flow. J. Differential Geom. 41: pp. 215226
 Chow, B, Chu, S (2001) Spacetime formulation of Harnack inequalities for curvature flows of hypersurfaces. J. Geom. Anal. 11: pp. 219231 CrossRef
 Huisken, G, Sinestrari, C (1999) Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math. 183: pp. 4570 CrossRef
 Caffarelli, L, Friedman, A (1985) Convexity of solutions of some semilinear elliptic equations. Duke Math. J. 52: pp. 431455 CrossRef
 Singer, I, Wong, B, Yau, S T (1985) An estimate of gap of the first two eigenvalues in the Schrodinger operator. Ann. Sc. Norm. Super. Pisa Cl. Sci. 12: pp. 319333
 Korevaar, N J, Lewis, J (1987) Convex solutions of certain elliptic equations have constant rank Hessians. Arch. Ration. Mech. Anal. 97: pp. 1932 CrossRef
 Bian, B J, Guan, P F (2009) A microscopic convexity principle for nonlinear partial differential equations. Invent. Math. 177: pp. 307335 CrossRef
 Bian, B J, Guan, P F (2010) A structural condition for microscopic convexity principle. Discrete Contin. Dyn. Syst. 28: pp. 789807 CrossRef
 Caffarelli, L, Guan, P F, Ma, X N (2007) A constant rank theorem for solutions of fully nonlinear elliptic equations. Comm. Pure Appl. Math. 60: pp. 17691791 CrossRef
 Guan, P F, Lin, C S, Ma, X N (2006) The ChristoffelMinkowski problem II: Weingarten curvature equations. Chin. Ann. Math. Ser. B 27: pp. 595614 CrossRef
 Guan, P F, Ma, X N (2003) The ChristoffelMinkowski problem I: Convexity of solutions of a Hessian equation. Invent. Math. 151: pp. 553577 CrossRef
 Guan, P F, Ma, X N, Zhou, F (2006) The ChristoffelMinkowski problem III: Existence and convexity of admissible solutions. Comm. Pure Appl. Math. 59: pp. 13521376 CrossRef
 Korevaar, N J (1990) Convexity of level sets for solutions to elliptic ring problems. Comm. Partial Differential Equations 15: pp. 541556 CrossRef
 Bian, B J, Guan, P F, Ma, X N (2011) A microscopic convexity principle for the level sets of solution for nonlinear elliptic partial diffarential equations. Indiana Univ. Math. J. 60: pp. 101120 CrossRef
 Bianchini, C, Longinetti, M, Salani, P (2009) Quasiconcave solutions to elliptic problems in convex rings. Indiana Univ. Math. J. 58: pp. 15651589 CrossRef
 Han, F, Ma, X N, Wu, D M (2011) The existence of kconvex hypersurface with prescribed mean curvature. Calc. Var. Partial Differential Equations 42: pp. 4372 CrossRef
 Liu, P, Ma, X N, Xu, L (2010) A BrunnMinkowski inequality for the Hessian eigenvalue in three dimension convex domain. Adv. Math. 225: pp. 16161633 CrossRef
 Ma, X N, Xu, L (2008) The convexity of solution of a class Hessian equation in bounded convex domain in ℝ3. J. Funct. Anal. 255: pp. 17131723 CrossRef
 Korevaar, N J (1983) Capillary surface convexity above convex domains. Indiana Univ. Math. J. 32: pp. 7381 CrossRef
 Korevaar, N J (1983) Convex solutions to nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J. 32: pp. 603614 CrossRef
 Kennington, A U (1985) Power concavity and boundary value problems. Indiana Univ. Math. J. 34: pp. 687704 CrossRef
 Kawohl, B (1986) A remark on N. Korevaars concavity maximum principle and on the asymptotic uniqueness of solutions to the plasma problem. Math. Methods Appl. Sci. 8: pp. 93101 CrossRef
 Alvarez, O, Lasry, J M, Lions, P L (1997) Convex viscosity solutions and state constraints. J. Math. Pures Appl. 76: pp. 265288
 Kennington, A U (1988) Convexity of level curves for an initial value problem. J. Math. Anal. Appl. 133: pp. 324330 CrossRef
 Porru, G, Serra, S (1994) Maximum principles for parabolic equations. J. Austral. Math. Soc. Ser. A 56: pp. 4152 CrossRef
 Hu, B W, Ma, X N (2012) Constant rank theorem of the spacetime convex solution of heat equation. Manu. Math. 138: pp. 89118 CrossRef
 Hu, B. W., Zhang, Y. B.: Spacetime convex solutions to the heat equation on manifolds. Preprint
 Lieberman, G (1996) Second Order Parabolic Differential Equations. World Scientific, Singapore CrossRef
 Zhu, X P (2002) Lectures on mean curvature flows. American Mathematical Society and International Press, Providence, RI
 Title
 A microscopic convexity principle for spacetime convex solutions of fully nonlinear parabolic equations
 Journal

Acta Mathematica Sinica, English Series
Volume 29, Issue 4 , pp 651674
 Cover Date
 20130401
 DOI
 10.1007/s101140121495z
 Print ISSN
 14398516
 Online ISSN
 14397617
 Publisher
 Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society
 Additional Links
 Topics
 Keywords

 Spacetime convexity
 microscopic convexity principle
 constant rank theorem
 parabolic convexity
 spacetime second fundamental form
 35K10
 35B99
 Authors

 Chuan Qiang Chen ^{(11495)}
 Bo Wen Hu ^{(21495)}
 Author Affiliations

 11495. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, P. R. China
 21495. Department of Mathematics and Applied Mathematics, Southwest University of Science and Technology, Mianyang, 621010, P. R. China