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Convexity of solutions to boundary blow-up problems
On behavior of signs for the heat equation and a diffusion method for data separation
1. | Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan |
2. | Graduate School of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Tokyo 153-8914 |
3. | Division of Mathematical Sciences, Graduate School of Engineering, Gunma University, 4-2 Aramaki-cho, Maebashi, 371-8510 |
4. | Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo, 153-8914 |
References:
[1] |
F. Andreu-Vaillo, V. Caselles and J. M. Mazón, "Parabolic Quasilinear Equations Minimizing Linear Growth Functionals,'' Progress in Mathematics, 223, Birkhäuser Verlag, 2004.
doi: 10.1007/978-3-0348-7928-6. |
[2] |
S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 79-96.
doi: 10.1515/crll.1988.390.79. |
[3] |
A. L. Bertozzi and A. Flenner, Diffuse interface models on graphs for classification of high dimensional data, Multiscale Modeling and Simulation, 10 (2012), 1090-1118.
doi: 10.1137/11083109X. |
[4] |
M. Bonforte and A. Figalli, Total variation flow and sign fast diffusion in one dimension, J. Differential Equations, 252 (2012), 4455-4480.
doi: 10.1016/j.jde.2012.01.003. |
[5] |
A. Briani, A. Chambolle, M. Novaga and G. Orlandi, On the gradient flow of a one-homogeneous functional, Confluentes Mathematici, 3 (2011), 617-635.
doi: 10.1142/S1793744211000461. |
[6] |
A. Chambolle, An algorithm for mean curvature motion, Interfaces and free boundaries, 6 (2004), 195-218.
doi: 10.4171/IFB/97. |
[7] |
X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141.
doi: 10.1016/0022-0396(92)90146-E. |
[8] |
X.-Y. Chen, A strong unique continuation theorem for parabolic equations, Math. Ann., 311 (1998), 603-630.
doi: 10.1007/s002080050202. |
[9] |
X.-Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. Differential Equations, 78 (1989), 160-190.
doi: 10.1016/0022-0396(89)90081-8. |
[10] |
R. R. Coifman, S. Lafon, A. B. Lee, M. Maggioni, B. Nadler, F. Warner and S. W. Zucker, Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps, Proc. Natl. Acad. Sci., 102 (2005), 7426-7431.
doi: 10.1073/pnas.0500334102. |
[11] |
N. Cristianini and J. Shawe-Taylor, "An Introduction to Support Vector Machines and Other Kernel-based Learning Methods,'' Cambridge University Press, 2000. |
[12] |
M.-H. Giga and Y. Giga, Very singular diffusion equations: second and fourth order problems, Japan J. Indust. Appl. Math., 27(2010), 323-345.
doi: 10.1007/s13160-010-0020-y. |
[13] |
M.-H. Giga, Y. Giga and J. Saal, "Nonlinear Partial Differential Equations: Asymptotic Behavior of Solutions and Self-similar Solutions,'' Progress in Nonlinear Differential Equations and Their Applications, 79, Birkhäuser, Boston, 2010.
doi: 10.1007/978-0-8176-4651-6. |
[14] |
D. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations, J. Differential Equations, 598 (1985), 165-205.
doi: 10.1016/0022-0396(85)90153-6. |
[15] |
K. Kielak, P. B. Mucha and P. Rybka, Almost classical solutions to the total variation flow. (to appear in Journal of Evolution Equations)
doi: 10.1007/s00028-012-0167-x. |
[16] |
H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), 401-441. |
[17] |
L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza: il Teorema di Modica-Mortola, Boll. Un. Mat. Ital., (B5) (1977), 285-299. |
[18] |
K. Nickel, Gestaltaussagen über Lösungen parabolischer Differentialgleighungen, J. Reine Angew. Math., 211 (1962), 78-94.
doi: 10.1515/crll.1962.211.78. |
[19] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'' Englewood Cliffs, 1967. |
[20] |
W. Ring, Structural properties of solutions to total variation regularization problems, M2AN Math. Model. Numer. Anal., 34 (2000), 799-811.
doi: 10.1051/m2an:2000104. |
[21] |
L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
[22] |
G. Steidl, Supervised learning by support vector machines, "Handbook of Mathematical Methods in Imaging'' (O. Scherzer ed.), vol 3, Springer, 2011, 959-1013. |
[23] |
P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal., 101 (1988), 209-260.
doi: 10.1007/BF00253122. |
[24] |
D. G. Widder, "The Heat Equations,'' Academic Press, New York, 1975. |
show all references
References:
[1] |
F. Andreu-Vaillo, V. Caselles and J. M. Mazón, "Parabolic Quasilinear Equations Minimizing Linear Growth Functionals,'' Progress in Mathematics, 223, Birkhäuser Verlag, 2004.
doi: 10.1007/978-3-0348-7928-6. |
[2] |
S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 79-96.
doi: 10.1515/crll.1988.390.79. |
[3] |
A. L. Bertozzi and A. Flenner, Diffuse interface models on graphs for classification of high dimensional data, Multiscale Modeling and Simulation, 10 (2012), 1090-1118.
doi: 10.1137/11083109X. |
[4] |
M. Bonforte and A. Figalli, Total variation flow and sign fast diffusion in one dimension, J. Differential Equations, 252 (2012), 4455-4480.
doi: 10.1016/j.jde.2012.01.003. |
[5] |
A. Briani, A. Chambolle, M. Novaga and G. Orlandi, On the gradient flow of a one-homogeneous functional, Confluentes Mathematici, 3 (2011), 617-635.
doi: 10.1142/S1793744211000461. |
[6] |
A. Chambolle, An algorithm for mean curvature motion, Interfaces and free boundaries, 6 (2004), 195-218.
doi: 10.4171/IFB/97. |
[7] |
X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141.
doi: 10.1016/0022-0396(92)90146-E. |
[8] |
X.-Y. Chen, A strong unique continuation theorem for parabolic equations, Math. Ann., 311 (1998), 603-630.
doi: 10.1007/s002080050202. |
[9] |
X.-Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. Differential Equations, 78 (1989), 160-190.
doi: 10.1016/0022-0396(89)90081-8. |
[10] |
R. R. Coifman, S. Lafon, A. B. Lee, M. Maggioni, B. Nadler, F. Warner and S. W. Zucker, Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps, Proc. Natl. Acad. Sci., 102 (2005), 7426-7431.
doi: 10.1073/pnas.0500334102. |
[11] |
N. Cristianini and J. Shawe-Taylor, "An Introduction to Support Vector Machines and Other Kernel-based Learning Methods,'' Cambridge University Press, 2000. |
[12] |
M.-H. Giga and Y. Giga, Very singular diffusion equations: second and fourth order problems, Japan J. Indust. Appl. Math., 27(2010), 323-345.
doi: 10.1007/s13160-010-0020-y. |
[13] |
M.-H. Giga, Y. Giga and J. Saal, "Nonlinear Partial Differential Equations: Asymptotic Behavior of Solutions and Self-similar Solutions,'' Progress in Nonlinear Differential Equations and Their Applications, 79, Birkhäuser, Boston, 2010.
doi: 10.1007/978-0-8176-4651-6. |
[14] |
D. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations, J. Differential Equations, 598 (1985), 165-205.
doi: 10.1016/0022-0396(85)90153-6. |
[15] |
K. Kielak, P. B. Mucha and P. Rybka, Almost classical solutions to the total variation flow. (to appear in Journal of Evolution Equations)
doi: 10.1007/s00028-012-0167-x. |
[16] |
H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), 401-441. |
[17] |
L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza: il Teorema di Modica-Mortola, Boll. Un. Mat. Ital., (B5) (1977), 285-299. |
[18] |
K. Nickel, Gestaltaussagen über Lösungen parabolischer Differentialgleighungen, J. Reine Angew. Math., 211 (1962), 78-94.
doi: 10.1515/crll.1962.211.78. |
[19] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'' Englewood Cliffs, 1967. |
[20] |
W. Ring, Structural properties of solutions to total variation regularization problems, M2AN Math. Model. Numer. Anal., 34 (2000), 799-811.
doi: 10.1051/m2an:2000104. |
[21] |
L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
[22] |
G. Steidl, Supervised learning by support vector machines, "Handbook of Mathematical Methods in Imaging'' (O. Scherzer ed.), vol 3, Springer, 2011, 959-1013. |
[23] |
P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal., 101 (1988), 209-260.
doi: 10.1007/BF00253122. |
[24] |
D. G. Widder, "The Heat Equations,'' Academic Press, New York, 1975. |
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