August  2011, 4(4): 851-864. doi: 10.3934/dcdss.2011.4.851

On a new kind of convexity for solutions of parabolic problems

1. 

Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578

2. 

Dipartimento di Matematica 'U. Dini', Viale Morgagni 67/A, 50137 Firenze, Italy

Received  December 2009 Revised  February 2010 Published  November 2010

We introduce the notion of $\alpha$-parabolic quasi-concavity for functions of space and time, which extends the usual notion of quasi-concavity and the notion of parabolic quasi-cocavity introduced in [18]. Then we investigate the $\alpha$-parabolic quasi-concavity of solutions to parabolic problems with vanishing initial datum. The results here obtained are generalizations of some of the results of [18].
Citation: Kazuhiro Ishige, Paolo Salani. On a new kind of convexity for solutions of parabolic problems. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 851-864. doi: 10.3934/dcdss.2011.4.851
References:
[1]

C. Bianchini, M. Longinetti and P. Salani, Quasiconcave solutions to elliptic problems in convex rings, Indiana Univ. Math J., 58 (2009), 1565-1590. doi: doi:10.1512/iumj.2009.58.3539.

[2]

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

[3]

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

[4]

H. J. Brascamp and E. H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Anal., 22 (1976), 366-389. doi: doi:10.1016/0022-1236(76)90004-5.

[5]

P. Daskalopoulos, R. Hamilton and K. Lee, All time $C^\infty$-Regularity of interface in degenerated diffusion: A geometric approach, Duke Math. Journal, 108 (2001), 295-327. doi: doi:10.1215/S0012-7094-01-10824-7.

[6]

P. Daskalopoulos and K.-A. Lee, Convexity and all-time $C^\infty$-regularity of the interface in flame propagation, Comm. Pure Appl. Math., 55 (2002), 633-653. doi: doi:10.1002/cpa.10028.

[7]

P. Daskalopoulos and K.-A. Lee, All time smooth solutions of the one-phase Stefan problem and the Hele-Shaw flow, Comm. in P.D.E., 12 (2004), 71-89.

[8]

J. I. Diaz and B. Kawohl, On convexity and starshapedness of level sets for some nonlinear elliptic and parabolic problems on convex rings, Preprint n. 393 (1986), Sonderforschungsbereich 123, Universität Heidelberg, available at http://www.mi.uni-koeln.de/ kawohl.

[9]

J. I. Diaz and B. Kawohl, On convexity and starshapedness of level sets for some nonlinear elliptic and parabolic problems on convex rings, J. Math. Anal. Appl., 177 (1993), 263-286. doi: doi:10.1006/jmaa.1993.1257.

[10]

E. Francini, Starshapedness of level sets for solutions of nonlinear parabolic equations, Rend. Ist. Mat. Univ. Trieste, 28 (1996), 49-62.

[11]

Y. Giga, S. Goto, H. Ishii and M.-H. Sato, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J., 40 (1991), 443-470. doi: doi:10.1512/iumj.1991.40.40023.

[12]

A. Greco, Extremality conditions for the quasi-concavity function and applications, Arch. Math., 93 (2009), 389-398. doi: doi:10.1007/s00013-009-0035-2.

[13]

A. Greco and B. Kawohl, Log-concavity in some parabolic problems, Electron. J. Differential Equations, 1999 (1999), 1-12.

[14]

P. Guan and Lu Xu, Extremality conditions for the quasi-concavity function and applications, eprint arXiv:1004.1187v2 (2010), http://arxiv.org/abs/1004.1187v2.

[15]

G. H. Hardy, J. E. Littlewood and G. Pólya, "Inequalities," Cambridge Univ. Press, 1934.

[16]

K. Ishige and P. Salani, Is quasi-concavity preserved by heat flow?, Arch. Math., 90 (2008), 455-460. doi: doi:10.1007/s00013-008-2437-y.

[17]

K. Ishige and P. Salani, Convexity breaking of free boundary in porous medium equation, Interfaces Free Bound., 12 (2010), 75-84. doi: doi:10.4171/IFB/227.

[18]

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

[19]

S. Janson and J. Tysk, Preservation of convexity of solutions to parabolic equations, J. Differential Equations, 206 (2004), 182-226. doi: doi:10.1016/j.jde.2004.07.016.

[20]

B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE," Lecture Notes in Math. 1150, Springer-Verlag, Berlin, 1985.

[21]

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

[22]

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

[23]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Amer. Math. Soc., Providence, 1968.

[24]

K.-A. Lee, Power-concavity on nonlinear parabolic flows, Comm. Pure Appl. Math., 58 (2005), 1529-1543. doi: doi:10.1002/cpa.20068.

[25]

K.-A. Lee and J. L. Vázquez, Geometrical properties of solutions of the porous medium equation for large times, Indiana Univ. Math. J., 52 (2003), 991-1016. doi: doi:10.1512/iumj.2003.52.2200.

[26]

P.-L. Lions and M. Musiela, Convexity of solutions of parabolic equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 915-921

[27]

M. Longinetti and P. Salani, On the Hessian matrix and Minkowski addition of quasiconcave functions, J. Math. Pures Appl., 88 (2007), 276-292. doi: doi:10.1016/j.matpur.2007.06.007.

show all references

References:
[1]

C. Bianchini, M. Longinetti and P. Salani, Quasiconcave solutions to elliptic problems in convex rings, Indiana Univ. Math J., 58 (2009), 1565-1590. doi: doi:10.1512/iumj.2009.58.3539.

[2]

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

[3]

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

[4]

H. J. Brascamp and E. H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Anal., 22 (1976), 366-389. doi: doi:10.1016/0022-1236(76)90004-5.

[5]

P. Daskalopoulos, R. Hamilton and K. Lee, All time $C^\infty$-Regularity of interface in degenerated diffusion: A geometric approach, Duke Math. Journal, 108 (2001), 295-327. doi: doi:10.1215/S0012-7094-01-10824-7.

[6]

P. Daskalopoulos and K.-A. Lee, Convexity and all-time $C^\infty$-regularity of the interface in flame propagation, Comm. Pure Appl. Math., 55 (2002), 633-653. doi: doi:10.1002/cpa.10028.

[7]

P. Daskalopoulos and K.-A. Lee, All time smooth solutions of the one-phase Stefan problem and the Hele-Shaw flow, Comm. in P.D.E., 12 (2004), 71-89.

[8]

J. I. Diaz and B. Kawohl, On convexity and starshapedness of level sets for some nonlinear elliptic and parabolic problems on convex rings, Preprint n. 393 (1986), Sonderforschungsbereich 123, Universität Heidelberg, available at http://www.mi.uni-koeln.de/ kawohl.

[9]

J. I. Diaz and B. Kawohl, On convexity and starshapedness of level sets for some nonlinear elliptic and parabolic problems on convex rings, J. Math. Anal. Appl., 177 (1993), 263-286. doi: doi:10.1006/jmaa.1993.1257.

[10]

E. Francini, Starshapedness of level sets for solutions of nonlinear parabolic equations, Rend. Ist. Mat. Univ. Trieste, 28 (1996), 49-62.

[11]

Y. Giga, S. Goto, H. Ishii and M.-H. Sato, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J., 40 (1991), 443-470. doi: doi:10.1512/iumj.1991.40.40023.

[12]

A. Greco, Extremality conditions for the quasi-concavity function and applications, Arch. Math., 93 (2009), 389-398. doi: doi:10.1007/s00013-009-0035-2.

[13]

A. Greco and B. Kawohl, Log-concavity in some parabolic problems, Electron. J. Differential Equations, 1999 (1999), 1-12.

[14]

P. Guan and Lu Xu, Extremality conditions for the quasi-concavity function and applications, eprint arXiv:1004.1187v2 (2010), http://arxiv.org/abs/1004.1187v2.

[15]

G. H. Hardy, J. E. Littlewood and G. Pólya, "Inequalities," Cambridge Univ. Press, 1934.

[16]

K. Ishige and P. Salani, Is quasi-concavity preserved by heat flow?, Arch. Math., 90 (2008), 455-460. doi: doi:10.1007/s00013-008-2437-y.

[17]

K. Ishige and P. Salani, Convexity breaking of free boundary in porous medium equation, Interfaces Free Bound., 12 (2010), 75-84. doi: doi:10.4171/IFB/227.

[18]

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

[19]

S. Janson and J. Tysk, Preservation of convexity of solutions to parabolic equations, J. Differential Equations, 206 (2004), 182-226. doi: doi:10.1016/j.jde.2004.07.016.

[20]

B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE," Lecture Notes in Math. 1150, Springer-Verlag, Berlin, 1985.

[21]

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

[22]

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

[23]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Amer. Math. Soc., Providence, 1968.

[24]

K.-A. Lee, Power-concavity on nonlinear parabolic flows, Comm. Pure Appl. Math., 58 (2005), 1529-1543. doi: doi:10.1002/cpa.20068.

[25]

K.-A. Lee and J. L. Vázquez, Geometrical properties of solutions of the porous medium equation for large times, Indiana Univ. Math. J., 52 (2003), 991-1016. doi: doi:10.1512/iumj.2003.52.2200.

[26]

P.-L. Lions and M. Musiela, Convexity of solutions of parabolic equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 915-921

[27]

M. Longinetti and P. Salani, On the Hessian matrix and Minkowski addition of quasiconcave functions, J. Math. Pures Appl., 88 (2007), 276-292. doi: doi:10.1016/j.matpur.2007.06.007.

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