Advanced Search
Article Contents
Article Contents

On the microscopic spacetime convexity principle for fully nonlinear parabolic equations II: Spacetime quasiconcave solutions

Abstract Related Papers Cited by
  • In [13], Chen-Ma-Salani established the strict convexity of spacetime level sets of solutions to heat equation in convex rings, using the constant rank theorem and a deformation method. In this paper, we generalize the constant rank theorem in [13] to fully nonlinear parabolic equations, that is, establish the corresponding microscopic spacetime convexity principles for spacetime level sets. In fact, the results hold for fully nonlinear parabolic equations under a general structural condition, including the $p$-Laplacian parabolic equations ($p >1$) and some mean curvature type parabolic equations.
    Mathematics Subject Classification: Primary: 35K10; Secondary: 35B99.


    \begin{equation} \\ \end{equation}
  • [1]

    L. V. Ahlfors, Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973.


    B. Bian and P. Guan, A microscopic convexity principle for nonlinear partial differential equations, Inventiones Math., 177 (2009), 307-335.doi: 10.1007/s00222-009-0179-5.


    B. Bian, P. Guan, X. N. Ma and L. Xu, A constant rank theorem for quasiconcave solutions of fully nonlinear partial differential equations, Indiana Univ. Math. J., 60 (2011), 101-119.doi: 10.1512/iumj.2011.60.4222.


    C. Bianchini, M. Longinetti and P. Salani, Quasiconcave solutions to elliptic problems in convex rings, Indiana Univ. Math. J., 58 (2009), 1565-1589.doi: 10.1512/iumj.2009.58.3539.


    C. Borell, Brownian motion in a convex ring and quasiconcavity, Comm. Math. Phys., 86 (1982), 143-147.doi: 10.1007/BF01205665.


    C. Borell, A note on parabolic convexity and heat conduction, Ann. Inst. H. Poincaré Probab. Statist., 32 (1996), 387-393.


    C. Borell, Diffusion equations and geometric inequalities, Potential Anal., 12 (2000), 49-71.doi: 10.1023/A:1008641618547.


    L. Caffarelli and A. Friedman, Convexity of solutions of some semilinear elliptic equations, Duke Math. J., 52 (1985), 431-456.doi: 10.1215/S0012-7094-85-05221-4.


    L. Caffarelli and J. Spruck, Convexity properties of solutions to some classical variational problems, Comm. Part. Diff. Eq., 7 (1982), 1337-1379.doi: 10.1080/03605308208820254.


    S.-Y. A. Chang, X. N. Ma and P. Yang, Principal curvature estimates for the convex level sets of semilinear elliptic equations, Discrete Contin. Dyn. Syst., 28 (2010), 1151-1164.doi: 10.3934/dcds.2010.28.1151.


    C. Q. Chen, On the microscopic spacetime convexity principle of fully nonlinear parabolic equations I: Spacetime convex solutions, Discrete Contin. Dyn. Syst. A, 34 (2014), 3383-3402.doi: 10.3934/dcds.2014.34.3383.


    C. Q. Chen and B. W. Hu, A microscopic convexity principle for spacetime convex solutions of fully nonlinear parabolic equations, Acta Mathematica Sinica, English Series, 29 (2013), 651-674.doi: 10.1007/s10114-012-1495-z.


    C. Q. Chen, X. N. Ma and P. Salani, On the spacetime quasiconcave solutions of the heat equation, preprint, arXiv:1405.6373.


    C. Q. Chen and S. J. Shi, Curvature estimates for the level sets of spatial quasiconcave solutions to a class of parabolic equations, Science China Mathematics, 54 (2011), 2063-2080.doi: 10.1007/s11425-011-4277-7.


    R. Gabriel, A result concerning convex level surfaces of 3-dimensional harmonic functions, J. London Math. Soc., 32 (1957), 286-294.


    P. Guan and L. Xu, Convexity estimates for level surfaces of quasiconcave solutions to fully nonlinear elliptic equations, J. Reine Angew. Math., 680 (2013), 41-67.doi: 10.1515/crelle.2012.038.


    B. W. Hu and X. N. Ma, Constant rank theorem of the spacetime convex solution of heat equation, manuscripta math., 138 (2012), 89-118.doi: 10.1007/s00229-011-0485-2.


    K. Ishige and P. Salani, Parabolic quasi-concavity for solutions to parabolic problems in convex rings, Math. Nachr., 283 (2010), 1526-1548.doi: 10.1002/mana.200910242.


    K. Ishige and P. Salani, On a new kind of convexity for solutions of parabolic problems, Discret. Contin. Dyn. Syst. Ser. S, 4 (2011), 851-864.doi: 10.3934/dcdss.2011.4.851.


    K. Ishige and P. Salani, Parabolic power concavity and parabolic boundary value problems, Math. Ann., 358 (2014), 1091-1117.doi: 10.1007/s00208-013-0991-5.


    B. Kawhol, Rearrangements and Convexity of Level Sets in PDE, Springer Lecture Notes in Math. 1150, 1985.


    N. Korevaar, Convexity of level sets for solutions to elliptic ring problems, Comm. Part. Diff. Eq., 15 (1990), 541-556.doi: 10.1080/03605309908820698.


    J. Lewis, Capacitary functions in convex rings, Arch. Rat. Mech. Anal., 66 (1977), 201-224.


    G. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1996.doi: 10.1142/3302.


    M. Longinetti, Convexity of the level lines of harmonic functions, (Italian) Boll. Un. Mat. Ital. A, 2 (1983), 71-75.


    M. Longinetti, On minimal surfaces bounded by two convex curves in parallel planes, J. Diff. Equations, 67 (1987), 344-358.doi: 10.1016/0022-0396(87)90131-8.


    X. N. Ma, Q. Z. Ou and W. Zhang, Gaussian curvature estimates for the convex level sets of $p$-harmonic functions, Comm. Pure Appl. Math., 63 (2010), 935-971.doi: 10.1002/cpa.20318.


    M. Ortel and W. Schneider, Curvature of level curves of harmonic functions, Canad. Math. Bull., 26 (1983), 399-405.doi: 10.4153/CMB-1983-066-4.


    M. Shiffman, On surfaces of stationary area bounded by two circles or convex curves in parallel planes, Annals of Math., 63 (1956), 77-90.doi: 10.2307/1969991.


    F. Treves, A new method of proof of the subelliptic estimates, Commun. Pure Appl. Math., 24 (1971), 71-115.doi: 10.1002/cpa.3160240107.


    L. Xu, A microscopic convexity theorem of level sets for solutions to elliptic equations, Cal. Var. PDE, 40 (2011), 51-63.doi: 10.1007/s00526-010-0333-3.

  • 加载中

Article Metrics

HTML views() PDF downloads(200) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint