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Hot spots for the two dimensional heat equation with a rapidly decaying negative potential
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 |
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. |
[2] |
C. Borell, Brownian motion in a convex ring and quasiconcavity,, Comm. Math. Phys., 86 (1982), 143.
doi: doi:10.1007/BF01205665. |
[3] |
C. Borell, A note on parabolic convexity and heat conduction,, Ann. Inst. H. Poincar\'e Probab. Statist., 32 (1996), 387.
|
[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. |
[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. |
[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. |
[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.
|
[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).
|
[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. |
[10] |
E. Francini, Starshapedness of level sets for solutions of nonlinear parabolic equations,, Rend. Ist. Mat. Univ. Trieste, 28 (1996), 49.
|
[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. |
[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. |
[13] |
A. Greco and B. Kawohl, Log-concavity in some parabolic problems,, Electron. J. Differential Equations, 1999 (1999), 1.
|
[14] |
P. Guan and Lu Xu, Extremality conditions for the quasi-concavity function and applications,, eprint arXiv:1004.1187v2 (2010), (2010).
|
[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.
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.
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.
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.
doi: doi:10.1016/j.jde.2004.07.016. |
[20] |
B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE,", Lecture Notes in Math. \textbf{1150}, 1150 (1985).
|
[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. |
[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. |
[23] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Amer. Math. Soc., (1968).
|
[24] |
K.-A. Lee, Power-concavity on nonlinear parabolic flows,, Comm. Pure Appl. Math., 58 (2005), 1529.
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.
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.
|
[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. |
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. |
[2] |
C. Borell, Brownian motion in a convex ring and quasiconcavity,, Comm. Math. Phys., 86 (1982), 143.
doi: doi:10.1007/BF01205665. |
[3] |
C. Borell, A note on parabolic convexity and heat conduction,, Ann. Inst. H. Poincar\'e Probab. Statist., 32 (1996), 387.
|
[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. |
[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. |
[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. |
[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.
|
[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).
|
[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. |
[10] |
E. Francini, Starshapedness of level sets for solutions of nonlinear parabolic equations,, Rend. Ist. Mat. Univ. Trieste, 28 (1996), 49.
|
[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. |
[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. |
[13] |
A. Greco and B. Kawohl, Log-concavity in some parabolic problems,, Electron. J. Differential Equations, 1999 (1999), 1.
|
[14] |
P. Guan and Lu Xu, Extremality conditions for the quasi-concavity function and applications,, eprint arXiv:1004.1187v2 (2010), (2010).
|
[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.
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.
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.
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.
doi: doi:10.1016/j.jde.2004.07.016. |
[20] |
B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE,", Lecture Notes in Math. \textbf{1150}, 1150 (1985).
|
[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. |
[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. |
[23] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Amer. Math. Soc., (1968).
|
[24] |
K.-A. Lee, Power-concavity on nonlinear parabolic flows,, Comm. Pure Appl. Math., 58 (2005), 1529.
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.
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.
|
[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. |
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