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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

Chuanqiang Chen 1,

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].
Keywords: fully nonlinear parabolic equation, constant rank theorem, Spacetime convexity, elementary symmetric functions., microscopic convexity principle.
Mathematics Subject Classification: Primary: 35K10; Secondary: 35B9.
Citation: Chuanqiang Chen. On the microscopic spacetime convexity principle of fully nonlinear parabolic equations I: Spacetime convex solutions. Discrete and Continuous Dynamical Systems, 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), 265-288. doi: 10.1016/S0021-7824(97)89952-7.

[2]

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

[3]

B. J. Bian and P. Guan, A structural condition for microscopic convexity principle, Discrete Contin. Dyn. Syst., 28 (2010), 789-807. doi: 10.3934/dcds.2010.28.789.

[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), 101-–119. doi: 10.1512/iumj.2011.60.4222.

[5]

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.

[6]

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

[7]

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

[8]

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

[9]

L. Caffarelli and A. Friedman, Convexity of solutions of some semilinear elliptic equations, Duke Math. J., 52 (1985), 281-545. doi: 10.1215/S0012-7094-85-05221-4.

[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), 1769-1791. doi: 10.1002/cpa.20197.

[11]

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.

[12]

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

[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), 595-614. doi: 10.1007/s11401-005-0575-0.

[14]

P. Guan and X. N. Ma, The Christoffel-Minkowski problem I: Convexity of solutions of a Hessian equation, Invent. Math., 151 (2003), 553-577. doi: 10.1007/s00222-002-0259-2.

[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), 1352-1376. doi: 10.1002/cpa.20118.

[16]

, P. Guan and X. W. Zhang, private communication.

[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), 43-72. doi: 10.1007/s00526-010-0379-2.

[18]

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.

[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), 93-101. doi: 10.1002/mma.1670080107.

[20]

A. U. Kennington, Power concavity and boundary value problems, Indiana Univ. Math. J., 34 (1985), 687-704. doi: 10.1512/iumj.1985.34.34036.

[21]

A. U. Kennington, Convexity of level curves for an initial value problem, J. Math. Anal. Appl., 133 (1988), 324-330. doi: 10.1016/0022-247X(88)90404-0.

[22]

N. J. Korevaar, Capillary surface convexity above convex domains, Indiana Univ. Math. J., 32 (1983), 73-81. doi: 10.1512/iumj.1983.32.32007.

[23]

N. J. Korevaar, Convex solutions to nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 32 (1983), 603-614. doi: 10.1512/iumj.1983.32.32042.

[24]

N. J. Korevaar, Convexity of level sets for solutions to elliptic ring problems, Comm. Partial Differential Equations, 15 (1990), 541-556. doi: 10.1080/03605309908820698.

[25]

N. J. Korevaar and J. Lewis, Convex solutions of certain elliptic equations have constant rank Hessians, Arch. Rational Mech. Anal., 97 (1987), 19-32. doi: 10.1007/BF00279844.

[26]

G. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1996.

[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), 1616-1633. doi: 10.1016/j.aim.2010.04.003.

[28]

X. N. Ma and L. Xu, The convexity of solution of a class Hessian equation in bounded convex domain in $\mathbb{R}^3$, J. Funct. Anal., 255 (2008), 1713-1723. doi: 10.1016/j.jfa.2008.06.008.

[29]

G. Porru and S. Serra, Maximum principles for parabolic equations, J. Austral. Math. Soc. Ser. A, 56 (1994), 41-52. doi: 10.1017/S1446788700034728.

[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), 319-333.

[31]

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

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), 265-288. doi: 10.1016/S0021-7824(97)89952-7.

[2]

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

[3]

B. J. Bian and P. Guan, A structural condition for microscopic convexity principle, Discrete Contin. Dyn. Syst., 28 (2010), 789-807. doi: 10.3934/dcds.2010.28.789.

[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), 101-–119. doi: 10.1512/iumj.2011.60.4222.

[5]

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.

[6]

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

[7]

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

[8]

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

[9]

L. Caffarelli and A. Friedman, Convexity of solutions of some semilinear elliptic equations, Duke Math. J., 52 (1985), 281-545. doi: 10.1215/S0012-7094-85-05221-4.

[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), 1769-1791. doi: 10.1002/cpa.20197.

[11]

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.

[12]

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

[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), 595-614. doi: 10.1007/s11401-005-0575-0.

[14]

P. Guan and X. N. Ma, The Christoffel-Minkowski problem I: Convexity of solutions of a Hessian equation, Invent. Math., 151 (2003), 553-577. doi: 10.1007/s00222-002-0259-2.

[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), 1352-1376. doi: 10.1002/cpa.20118.

[16]

, P. Guan and X. W. Zhang, private communication.

[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), 43-72. doi: 10.1007/s00526-010-0379-2.

[18]

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.

[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), 93-101. doi: 10.1002/mma.1670080107.

[20]

A. U. Kennington, Power concavity and boundary value problems, Indiana Univ. Math. J., 34 (1985), 687-704. doi: 10.1512/iumj.1985.34.34036.

[21]

A. U. Kennington, Convexity of level curves for an initial value problem, J. Math. Anal. Appl., 133 (1988), 324-330. doi: 10.1016/0022-247X(88)90404-0.

[22]

N. J. Korevaar, Capillary surface convexity above convex domains, Indiana Univ. Math. J., 32 (1983), 73-81. doi: 10.1512/iumj.1983.32.32007.

[23]

N. J. Korevaar, Convex solutions to nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 32 (1983), 603-614. doi: 10.1512/iumj.1983.32.32042.

[24]

N. J. Korevaar, Convexity of level sets for solutions to elliptic ring problems, Comm. Partial Differential Equations, 15 (1990), 541-556. doi: 10.1080/03605309908820698.

[25]

N. J. Korevaar and J. Lewis, Convex solutions of certain elliptic equations have constant rank Hessians, Arch. Rational Mech. Anal., 97 (1987), 19-32. doi: 10.1007/BF00279844.

[26]

G. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1996.

[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), 1616-1633. doi: 10.1016/j.aim.2010.04.003.

[28]

X. N. Ma and L. Xu, The convexity of solution of a class Hessian equation in bounded convex domain in $\mathbb{R}^3$, J. Funct. Anal., 255 (2008), 1713-1723. doi: 10.1016/j.jfa.2008.06.008.

[29]

G. Porru and S. Serra, Maximum principles for parabolic equations, J. Austral. Math. Soc. Ser. A, 56 (1994), 41-52. doi: 10.1017/S1446788700034728.

[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), 319-333.

[31]

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

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