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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 & Continuous Dynamical Systems, 2014, 34 (9) : 3383-3402. doi: 10.3934/dcds.2014.34.3383
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show all references

References:
[1]

J. Math. Pures Appl., 76 (1997), 265-288. doi: 10.1016/S0021-7824(97)89952-7.  Google Scholar

[2]

Invent. Math., 177 (2009), 307-335. doi: 10.1007/s00222-009-0179-5.  Google Scholar

[3]

Discrete Contin. Dyn. Syst., 28 (2010), 789-807. doi: 10.3934/dcds.2010.28.789.  Google Scholar

[4]

Indiana Univ. Math. J., 60 (2011), 101-–119. doi: 10.1512/iumj.2011.60.4222.  Google Scholar

[5]

Indiana Univ. Math. J., 58 (2009), 1565-1589. doi: 10.1512/iumj.2009.58.3539.  Google Scholar

[6]

Comm. Math. Phys., 86 (1982), 143-147. doi: 10.1007/BF01205665.  Google Scholar

[7]

Ann. Inst. H. Poincaré Probab. Statist., 32 (1996), 387-393.  Google Scholar

[8]

Potential Anal., 12 (2000), 49-71. doi: 10.1023/A:1008641618547.  Google Scholar

[9]

Duke Math. J., 52 (1985), 281-545. doi: 10.1215/S0012-7094-85-05221-4.  Google Scholar

[10]

Comm. Pure Appl. Math., 60 (2007), 1769-1791. doi: 10.1002/cpa.20197.  Google Scholar

[11]

Acta Mathematica Sinica, English Series, 29 (2013), 651-674. 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]

Chinese Ann. Math. Ser. B, 27 (2006), 595-614. doi: 10.1007/s11401-005-0575-0.  Google Scholar

[14]

Invent. Math., 151 (2003), 553-577. doi: 10.1007/s00222-002-0259-2.  Google Scholar

[15]

Comm. Pure Appl. Math., 59 (2006), 1352-1376. doi: 10.1002/cpa.20118.  Google Scholar

[16]

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

[17]

Calc. Var. Partial Differential Equations, 42 (2011), 43-72. doi: 10.1007/s00526-010-0379-2.  Google Scholar

[18]

Manuscripta Math., 138 (2012), 89-118. doi: 10.1007/s00229-011-0485-2.  Google Scholar

[19]

Math. Methods Appl. Sci., 8 (1986), 93-101. doi: 10.1002/mma.1670080107.  Google Scholar

[20]

Indiana Univ. Math. J., 34 (1985), 687-704. doi: 10.1512/iumj.1985.34.34036.  Google Scholar

[21]

J. Math. Anal. Appl., 133 (1988), 324-330. doi: 10.1016/0022-247X(88)90404-0.  Google Scholar

[22]

Indiana Univ. Math. J., 32 (1983), 73-81. doi: 10.1512/iumj.1983.32.32007.  Google Scholar

[23]

Indiana Univ. Math. J., 32 (1983), 603-614. doi: 10.1512/iumj.1983.32.32042.  Google Scholar

[24]

Comm. Partial Differential Equations, 15 (1990), 541-556. doi: 10.1080/03605309908820698.  Google Scholar

[25]

Arch. Rational Mech. Anal., 97 (1987), 19-32. doi: 10.1007/BF00279844.  Google Scholar

[26]

World Scientific, 1996.  Google Scholar

[27]

Adv. Math., 225 (2010), 1616-1633. doi: 10.1016/j.aim.2010.04.003.  Google Scholar

[28]

J. Funct. Anal., 255 (2008), 1713-1723. doi: 10.1016/j.jfa.2008.06.008.  Google Scholar

[29]

J. Austral. Math. Soc. Ser. A, 56 (1994), 41-52. doi: 10.1017/S1446788700034728.  Google Scholar

[30]

Ann. Scuola Norm. Sup. Pisa Cl. Sci., 12 (1985), 319-333.  Google Scholar

[31]

Commun. Pure Appl. Math., 24 (1971), 71-115. doi: 10.1002/cpa.3160240107.  Google Scholar

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