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September  2014, 34(9): 3383-3402. doi: 10.3934/dcds.2014.34.3383

On the microscopic spacetime convexity principle of fully nonlinear parabolic equations I: Spacetime convex solutions

1. 

Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China, Hefei, 230026, Anhui Province, China

Received  August 2013 Revised  December 2013 Published  March 2014

Spacetime convexity is a basic geometric property of the solutions of parabolic equations. In this paper, we study microscopic convexity properties of spacetime convex solutions of fully nonlinear parabolic partial differential equations and give a new simple proof of the constant rank theorem in [11].
Citation: Chuanqiang Chen. On the microscopic spacetime convexity principle of fully nonlinear parabolic equations I: Spacetime convex solutions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3383-3402. doi: 10.3934/dcds.2014.34.3383
References:
[1]

O. Alvarez, J. M. Lasry and P. L. Lions, Convex viscosity solutions and state constraints,, J. Math. Pures Appl., 76 (1997), 265.  doi: 10.1016/S0021-7824(97)89952-7.  Google Scholar

[2]

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

[3]

B. J. Bian and P. Guan, A structural condition for microscopic convexity principle,, Discrete Contin. Dyn. Syst., 28 (2010), 789.  doi: 10.3934/dcds.2010.28.789.  Google Scholar

[4]

B. J. 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).  doi: 10.1512/iumj.2011.60.4222.  Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

L. Caffarelli and A. Friedman, Convexity of solutions of some semilinear elliptic equations,, Duke Math. J., 52 (1985), 281.  doi: 10.1215/S0012-7094-85-05221-4.  Google Scholar

[10]

L. Caffarelli, P. Guan and X. N. Ma, A constant rank theorem for solutions of fully nonlinear elliptic equations,, Comm. Pure Appl. Math., 60 (2007), 1769.  doi: 10.1002/cpa.20197.  Google Scholar

[11]

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

[12]

C. Q. Chen, X. N. Ma and P. Salani, On the spacetime quasiconcave solutions of the heat equation,, preprint., ().   Google Scholar

[13]

P. Guan, C. S. Lin and X. N. Ma, The Christoffel-Minkowski problem II: Weingarten curvature equations,, Chinese Ann. Math. Ser. B, 27 (2006), 595.  doi: 10.1007/s11401-005-0575-0.  Google Scholar

[14]

P. Guan and X. N. Ma, The Christoffel-Minkowski problem I: Convexity of solutions of a Hessian equation,, Invent. Math., 151 (2003), 553.  doi: 10.1007/s00222-002-0259-2.  Google Scholar

[15]

P. Guan, X. N. Ma and F. Zhou, The Christoffel-Minkowski problem III: Existence and convexity of admissible solutions,, Comm. Pure Appl. Math., 59 (2006), 1352.  doi: 10.1002/cpa.20118.  Google Scholar

[16]

, P. Guan and X. W. Zhang,, private communication., ().   Google Scholar

[17]

F. Han, X. N. Ma and D. M. Wu, The existence of $k$-convex hypersurface with prescribed mean curvature,, Calc. Var. Partial Differential Equations, 42 (2011), 43.  doi: 10.1007/s00526-010-0379-2.  Google Scholar

[18]

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

[19]

B. Kawohl, A remark on N.Korevaar's concavity maximum principle and on the asymptotic uniqueness of solutions to the plasma problem,, Math. Methods Appl. Sci., 8 (1986), 93.  doi: 10.1002/mma.1670080107.  Google Scholar

[20]

A. U. Kennington, Power concavity and boundary value problems,, Indiana Univ. Math. J., 34 (1985), 687.  doi: 10.1512/iumj.1985.34.34036.  Google Scholar

[21]

A. U. Kennington, Convexity of level curves for an initial value problem,, J. Math. Anal. Appl., 133 (1988), 324.  doi: 10.1016/0022-247X(88)90404-0.  Google Scholar

[22]

N. J. Korevaar, Capillary surface convexity above convex domains,, Indiana Univ. Math. J., 32 (1983), 73.  doi: 10.1512/iumj.1983.32.32007.  Google Scholar

[23]

N. J. Korevaar, Convex solutions to nonlinear elliptic and parabolic boundary value problems,, Indiana Univ. Math. J., 32 (1983), 603.  doi: 10.1512/iumj.1983.32.32042.  Google Scholar

[24]

N. J. Korevaar, Convexity of level sets for solutions to elliptic ring problems,, Comm. Partial Differential Equations, 15 (1990), 541.  doi: 10.1080/03605309908820698.  Google Scholar

[25]

N. J. Korevaar and J. Lewis, Convex solutions of certain elliptic equations have constant rank Hessians,, Arch. Rational Mech. Anal., 97 (1987), 19.  doi: 10.1007/BF00279844.  Google Scholar

[26]

G. Lieberman, Second Order Parabolic Differential Equations,, World Scientific, (1996).   Google Scholar

[27]

P. Liu, X. N. Ma and L. Xu, A Brunn-Minkowski inequality for the Hessian eigenvalue in three dimension convex domain,, Adv. Math., 225 (2010), 1616.  doi: 10.1016/j.aim.2010.04.003.  Google Scholar

[28]

X. N. Ma and L. Xu, The convexity of solution of a class Hessian equation in bounded convex domain in $\mathbbR^3$,, J. Funct. Anal., 255 (2008), 1713.  doi: 10.1016/j.jfa.2008.06.008.  Google Scholar

[29]

G. Porru and S. Serra, Maximum principles for parabolic equations,, J. Austral. Math. Soc. Ser. A, 56 (1994), 41.  doi: 10.1017/S1446788700034728.  Google Scholar

[30]

I. Singer, B. Wong, S. T. Yau and S. S. T. Yau, An estimate of gap of the first two eigenvalues in the Schrodinger operator,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 12 (1985), 319.   Google Scholar

[31]

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

show all references

References:
[1]

O. Alvarez, J. M. Lasry and P. L. Lions, Convex viscosity solutions and state constraints,, J. Math. Pures Appl., 76 (1997), 265.  doi: 10.1016/S0021-7824(97)89952-7.  Google Scholar

[2]

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

[3]

B. J. Bian and P. Guan, A structural condition for microscopic convexity principle,, Discrete Contin. Dyn. Syst., 28 (2010), 789.  doi: 10.3934/dcds.2010.28.789.  Google Scholar

[4]

B. J. 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).  doi: 10.1512/iumj.2011.60.4222.  Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

L. Caffarelli and A. Friedman, Convexity of solutions of some semilinear elliptic equations,, Duke Math. J., 52 (1985), 281.  doi: 10.1215/S0012-7094-85-05221-4.  Google Scholar

[10]

L. Caffarelli, P. Guan and X. N. Ma, A constant rank theorem for solutions of fully nonlinear elliptic equations,, Comm. Pure Appl. Math., 60 (2007), 1769.  doi: 10.1002/cpa.20197.  Google Scholar

[11]

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

[12]

C. Q. Chen, X. N. Ma and P. Salani, On the spacetime quasiconcave solutions of the heat equation,, preprint., ().   Google Scholar

[13]

P. Guan, C. S. Lin and X. N. Ma, The Christoffel-Minkowski problem II: Weingarten curvature equations,, Chinese Ann. Math. Ser. B, 27 (2006), 595.  doi: 10.1007/s11401-005-0575-0.  Google Scholar

[14]

P. Guan and X. N. Ma, The Christoffel-Minkowski problem I: Convexity of solutions of a Hessian equation,, Invent. Math., 151 (2003), 553.  doi: 10.1007/s00222-002-0259-2.  Google Scholar

[15]

P. Guan, X. N. Ma and F. Zhou, The Christoffel-Minkowski problem III: Existence and convexity of admissible solutions,, Comm. Pure Appl. Math., 59 (2006), 1352.  doi: 10.1002/cpa.20118.  Google Scholar

[16]

, P. Guan and X. W. Zhang,, private communication., ().   Google Scholar

[17]

F. Han, X. N. Ma and D. M. Wu, The existence of $k$-convex hypersurface with prescribed mean curvature,, Calc. Var. Partial Differential Equations, 42 (2011), 43.  doi: 10.1007/s00526-010-0379-2.  Google Scholar

[18]

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

[19]

B. Kawohl, A remark on N.Korevaar's concavity maximum principle and on the asymptotic uniqueness of solutions to the plasma problem,, Math. Methods Appl. Sci., 8 (1986), 93.  doi: 10.1002/mma.1670080107.  Google Scholar

[20]

A. U. Kennington, Power concavity and boundary value problems,, Indiana Univ. Math. J., 34 (1985), 687.  doi: 10.1512/iumj.1985.34.34036.  Google Scholar

[21]

A. U. Kennington, Convexity of level curves for an initial value problem,, J. Math. Anal. Appl., 133 (1988), 324.  doi: 10.1016/0022-247X(88)90404-0.  Google Scholar

[22]

N. J. Korevaar, Capillary surface convexity above convex domains,, Indiana Univ. Math. J., 32 (1983), 73.  doi: 10.1512/iumj.1983.32.32007.  Google Scholar

[23]

N. J. Korevaar, Convex solutions to nonlinear elliptic and parabolic boundary value problems,, Indiana Univ. Math. J., 32 (1983), 603.  doi: 10.1512/iumj.1983.32.32042.  Google Scholar

[24]

N. J. Korevaar, Convexity of level sets for solutions to elliptic ring problems,, Comm. Partial Differential Equations, 15 (1990), 541.  doi: 10.1080/03605309908820698.  Google Scholar

[25]

N. J. Korevaar and J. Lewis, Convex solutions of certain elliptic equations have constant rank Hessians,, Arch. Rational Mech. Anal., 97 (1987), 19.  doi: 10.1007/BF00279844.  Google Scholar

[26]

G. Lieberman, Second Order Parabolic Differential Equations,, World Scientific, (1996).   Google Scholar

[27]

P. Liu, X. N. Ma and L. Xu, A Brunn-Minkowski inequality for the Hessian eigenvalue in three dimension convex domain,, Adv. Math., 225 (2010), 1616.  doi: 10.1016/j.aim.2010.04.003.  Google Scholar

[28]

X. N. Ma and L. Xu, The convexity of solution of a class Hessian equation in bounded convex domain in $\mathbbR^3$,, J. Funct. Anal., 255 (2008), 1713.  doi: 10.1016/j.jfa.2008.06.008.  Google Scholar

[29]

G. Porru and S. Serra, Maximum principles for parabolic equations,, J. Austral. Math. Soc. Ser. A, 56 (1994), 41.  doi: 10.1017/S1446788700034728.  Google Scholar

[30]

I. Singer, B. Wong, S. T. Yau and S. S. T. Yau, An estimate of gap of the first two eigenvalues in the Schrodinger operator,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 12 (1985), 319.   Google Scholar

[31]

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

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