# American Institute of Mathematical Sciences

September  2016, 36(9): 4761-4811. doi: 10.3934/dcds.2016007

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

 1 Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou, 310023, Zhejiang Province, China

Received  November 2014 Revised  March 2016 Published  May 2016

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.
Citation: Chuanqiang Chen. On the microscopic spacetime convexity principle for fully nonlinear parabolic equations II: Spacetime quasiconcave solutions. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4761-4811. doi: 10.3934/dcds.2016007
##### References:
 [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.  Google Scholar [2] 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.  Google Scholar [3] 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.  Google Scholar [4] 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.  Google Scholar [5] C. Borell, Brownian motion in a convex ring and quasiconcavity, Comm. Math. Phys., 86 (1982), 143-147. doi: 10.1007/BF01205665.  Google Scholar [6] C. Borell, A note on parabolic convexity and heat conduction, Ann. Inst. H. Poincaré Probab. Statist., 32 (1996), 387-393.  Google Scholar [7] C. Borell, Diffusion equations and geometric inequalities, Potential Anal., 12 (2000), 49-71. doi: 10.1023/A:1008641618547.  Google Scholar [8] 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.  Google Scholar [9] 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.  Google Scholar [10] 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.  Google Scholar [11] 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.  Google Scholar [12] 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.  Google Scholar [13] C. Q. Chen, X. N. Ma and P. Salani, On the spacetime quasiconcave solutions of the heat equation,, preprint, ().   Google Scholar [14] 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.  Google Scholar [15] R. Gabriel, A result concerning convex level surfaces of 3-dimensional harmonic functions, J. London Math. Soc., 32 (1957), 286-294.  Google Scholar [16] 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.  Google Scholar [17] 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.  Google Scholar [18] 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.  Google Scholar [19] 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.  Google Scholar [20] 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.  Google Scholar [21] B. Kawhol, Rearrangements and Convexity of Level Sets in PDE, Springer Lecture Notes in Math. 1150, 1985.  Google Scholar [22] N. Korevaar, Convexity of level sets for solutions to elliptic ring problems, Comm. Part. Diff. Eq., 15 (1990), 541-556. doi: 10.1080/03605309908820698.  Google Scholar [23] J. Lewis, Capacitary functions in convex rings, Arch. Rat. Mech. Anal., 66 (1977), 201-224.  Google Scholar [24] G. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1996. doi: 10.1142/3302.  Google Scholar [25] M. Longinetti, Convexity of the level lines of harmonic functions, (Italian) Boll. Un. Mat. Ital. A, 2 (1983), 71-75.  Google Scholar [26] 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.  Google Scholar [27] 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.  Google Scholar [28] 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.  Google Scholar [29] 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.  Google Scholar [30] F. Treves, A new method of proof of the subelliptic estimates, Commun. Pure Appl. Math., 24 (1971), 71-115. doi: 10.1002/cpa.3160240107.  Google Scholar [31] 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.  Google Scholar

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##### References:
 [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.  Google Scholar [2] 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.  Google Scholar [3] 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.  Google Scholar [4] 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.  Google Scholar [5] C. Borell, Brownian motion in a convex ring and quasiconcavity, Comm. Math. Phys., 86 (1982), 143-147. doi: 10.1007/BF01205665.  Google Scholar [6] C. Borell, A note on parabolic convexity and heat conduction, Ann. Inst. H. Poincaré Probab. Statist., 32 (1996), 387-393.  Google Scholar [7] C. Borell, Diffusion equations and geometric inequalities, Potential Anal., 12 (2000), 49-71. doi: 10.1023/A:1008641618547.  Google Scholar [8] 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.  Google Scholar [9] 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.  Google Scholar [10] 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.  Google Scholar [11] 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.  Google Scholar [12] 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.  Google Scholar [13] C. Q. Chen, X. N. Ma and P. Salani, On the spacetime quasiconcave solutions of the heat equation,, preprint, ().   Google Scholar [14] 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.  Google Scholar [15] R. Gabriel, A result concerning convex level surfaces of 3-dimensional harmonic functions, J. London Math. Soc., 32 (1957), 286-294.  Google Scholar [16] 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.  Google Scholar [17] 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.  Google Scholar [18] 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.  Google Scholar [19] 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.  Google Scholar [20] 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.  Google Scholar [21] B. Kawhol, Rearrangements and Convexity of Level Sets in PDE, Springer Lecture Notes in Math. 1150, 1985.  Google Scholar [22] N. Korevaar, Convexity of level sets for solutions to elliptic ring problems, Comm. Part. Diff. Eq., 15 (1990), 541-556. doi: 10.1080/03605309908820698.  Google Scholar [23] J. Lewis, Capacitary functions in convex rings, Arch. Rat. Mech. Anal., 66 (1977), 201-224.  Google Scholar [24] G. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1996. doi: 10.1142/3302.  Google Scholar [25] M. Longinetti, Convexity of the level lines of harmonic functions, (Italian) Boll. Un. Mat. Ital. A, 2 (1983), 71-75.  Google Scholar [26] 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.  Google Scholar [27] 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.  Google Scholar [28] 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.  Google Scholar [29] 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.  Google Scholar [30] F. Treves, A new method of proof of the subelliptic estimates, Commun. Pure Appl. Math., 24 (1971), 71-115. doi: 10.1002/cpa.3160240107.  Google Scholar [31] 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.  Google Scholar
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