American Institute of Mathematical Sciences

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

On a new kind of convexity for solutions of parabolic problems

Kazuhiro Ishige 1, and  Paolo Salani 2,

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].
Keywords: Parabolic equations, convexity..
Mathematics Subject Classification: Primary: 35K20; Secondary: 52A2.
Citation: Kazuhiro Ishige, Paolo Salani. On a new kind of convexity for solutions of parabolic problems. Discrete & 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.  doi: doi:10.1512/iumj.2009.58.3539.  Google Scholar

[2]

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

[3]

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

[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.  doi: doi:10.1016/0022-1236(76)90004-5.  Google Scholar

[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.  doi: doi:10.1215/S0012-7094-01-10824-7.  Google Scholar

[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.  doi: doi:10.1002/cpa.10028.  Google Scholar

[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.   Google Scholar

[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), 123 (1986).   Google Scholar

[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.  doi: doi:10.1006/jmaa.1993.1257.  Google Scholar

[10]

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

[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.  doi: doi:10.1512/iumj.1991.40.40023.  Google Scholar

[12]

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

[13]

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

[14]

P. Guan and Lu Xu, Extremality conditions for the quasi-concavity function and applications,, eprint arXiv:1004.1187v2 (2010), (2010).   Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE,", Lecture Notes in Math. \textbf{1150}, 1150 (1985).   Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[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.  doi: doi:10.1512/iumj.2003.52.2200.  Google Scholar

[26]

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

[27]

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

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.  doi: doi:10.1512/iumj.2009.58.3539.  Google Scholar

[2]

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

[3]

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

[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.  doi: doi:10.1016/0022-1236(76)90004-5.  Google Scholar

[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.  doi: doi:10.1215/S0012-7094-01-10824-7.  Google Scholar

[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.  doi: doi:10.1002/cpa.10028.  Google Scholar

[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.   Google Scholar

[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), 123 (1986).   Google Scholar

[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.  doi: doi:10.1006/jmaa.1993.1257.  Google Scholar

[10]

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

[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.  doi: doi:10.1512/iumj.1991.40.40023.  Google Scholar

[12]

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

[13]

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

[14]

P. Guan and Lu Xu, Extremality conditions for the quasi-concavity function and applications,, eprint arXiv:1004.1187v2 (2010), (2010).   Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE,", Lecture Notes in Math. \textbf{1150}, 1150 (1985).   Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[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.  doi: doi:10.1512/iumj.2003.52.2200.  Google Scholar

[26]

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

[27]

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

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