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 & 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.  Google Scholar

[2]

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

[3]

C. Borell, A note on parabolic convexity and heat conduction, Ann. Inst. H. Poincaré Probab. Statist., 32 (1996), 387-393.  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), 366-389. 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), 295-327. 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), 633-653. 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), 71-89.  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), Sonderforschungsbereich 123, Universität Heidelberg, available at http://www.mi.uni-koeln.de/ kawohl.  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), 263-286. 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), 49-62.  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), 443-470. 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), 389-398. 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), 1-12.  Google Scholar

[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.  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), 455-460. 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), 75-84. 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), 1526-1548. 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), 182-226. 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. 1150, Springer-Verlag, Berlin, 1985.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

K.-A. Lee, Power-concavity on nonlinear parabolic flows, Comm. Pure Appl. Math., 58 (2005), 1529-1543. 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), 991-1016. 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), 915-921  Google Scholar

[27]

M. Longinetti and P. Salani, On the Hessian matrix and Minkowski addition of quasiconcave functions, J. Math. Pures Appl., 88 (2007), 276-292. 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), 1565-1590. 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), 143-147. doi: doi:10.1007/BF01205665.  Google Scholar

[3]

C. Borell, A note on parabolic convexity and heat conduction, Ann. Inst. H. Poincaré Probab. Statist., 32 (1996), 387-393.  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), 366-389. 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), 295-327. 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), 633-653. 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), 71-89.  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), Sonderforschungsbereich 123, Universität Heidelberg, available at http://www.mi.uni-koeln.de/ kawohl.  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), 263-286. 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), 49-62.  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), 443-470. 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), 389-398. 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), 1-12.  Google Scholar

[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.  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), 455-460. 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), 75-84. 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), 1526-1548. 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), 182-226. 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. 1150, Springer-Verlag, Berlin, 1985.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

K.-A. Lee, Power-concavity on nonlinear parabolic flows, Comm. Pure Appl. Math., 58 (2005), 1529-1543. 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), 991-1016. 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), 915-921  Google Scholar

[27]

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

[1]

Kim Dang Phung. Carleman commutator approach in logarithmic convexity for parabolic equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 899-933. doi: 10.3934/mcrf.2018040

[2]

Chuanqiang Chen. On the microscopic spacetime convexity principle for fully nonlinear parabolic equations II: Spacetime quasiconcave solutions. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4761-4811. doi: 10.3934/dcds.2016007

[3]

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

[4]

David L. Finn. Convexity of level curves for solutions to semilinear elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1335-1343. doi: 10.3934/cpaa.2008.7.1335

[5]

Qing Liu, Atsushi Nakayasu. Convexity preserving properties for Hamilton-Jacobi equations in geodesic spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 157-183. doi: 10.3934/dcds.2019007

[6]

Juan Carlos Marrero, David Martín de Diego, Eduardo Martínez. Local convexity for second order differential equations on a Lie algebroid. Journal of Geometric Mechanics, 2021, 13 (3) : 477-499. doi: 10.3934/jgm.2021021

[7]

Victor Isakov. On increasing stability of the continuation for elliptic equations of second order without (pseudo)convexity assumptions. Inverse Problems & Imaging, 2019, 13 (5) : 983-1006. doi: 10.3934/ipi.2019044

[8]

Juraj Földes, Peter Poláčik. On asymptotically symmetric parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 673-689. doi: 10.3934/nhm.2012.7.673

[9]

H. Gajewski, I. V. Skrypnik. To the uniqueness problem for nonlinear parabolic equations. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 315-336. doi: 10.3934/dcds.2004.10.315

[10]

Jan Prüss, Gieri Simonett, Rico Zacher. On normal stability for nonlinear parabolic equations. Conference Publications, 2009, 2009 (Special) : 612-621. doi: 10.3934/proc.2009.2009.612

[11]

Wolfgang Walter. Nonlinear parabolic differential equations and inequalities. Discrete & Continuous Dynamical Systems, 2002, 8 (2) : 451-468. doi: 10.3934/dcds.2002.8.451

[12]

Guangying Lv, Hongjun Gao, Jinlong Wei, Jiang-Lun Wu. The effect of noise intensity on parabolic equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1715-1728. doi: 10.3934/dcdsb.2019248

[13]

Farid Ammar Khodja, Cherif Bouzidi, Cédric Dupaix, Lahcen Maniar. Null controllability of retarded parabolic equations. Mathematical Control & Related Fields, 2014, 4 (1) : 1-15. doi: 10.3934/mcrf.2014.4.1

[14]

Nikolaos Roidos. Expanding solutions of quasilinear parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1413-1429. doi: 10.3934/cpaa.2021026

[15]

Christopher Goodrich, Carlos Lizama. Positivity, monotonicity, and convexity for convolution operators. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 4961-4983. doi: 10.3934/dcds.2020207

[16]

Baojun Bian, Pengfei Guan. A structural condition for microscopic convexity principle. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 789-807. doi: 10.3934/dcds.2010.28.789

[17]

Arrigo Cellina, Carlo Mariconda, Giulia Treu. Comparison results without strict convexity. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 57-65. doi: 10.3934/dcdsb.2009.11.57

[18]

Flank D. M. Bezerra, Jacson Simsen, Mariza Stefanello Simsen. Convergence of quasilinear parabolic equations to semilinear equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3823-3834. doi: 10.3934/dcdsb.2020258

[19]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327

[20]

Takesi Fukao, Masahiro Kubo. Nonlinear degenerate parabolic equations for a thermohydraulic model. Conference Publications, 2007, 2007 (Special) : 399-408. doi: 10.3934/proc.2007.2007.399

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (108)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]