CNA-PIRE Mathematics Working
Group
Variational methods for phase transitions and copolymers
Tuesdays, 2:30pm - 3:30pm
Wean Hall 7218
We will investigate the application of variational methods to the study and understanding of phase transitions in materials and phase separation in copolymers.
Resources
[1] Alberti, Giovanni; Choksi, Rustum; Otto, Felix Uniform energy distribution for an isoperimetric problem with long-range interactions. J. Amer. Math. Soc. 22 (2009), no. 2, 569-605.
[2] Chermisi, M.; Dal Maso, G.; Fonseca, I.; Leoni, G. Singuar perturbation models in phase transitions for second order materials (2010).
[3] Choksi, Rustum; Peletier, Mark Small volume fraction limit of the diblock copolymer problem: I. Sharp-interface functional. SIAM J. Math. Anal. 42 (2010), no. 3, 1334-1370.
[4] Choksi, Rustum; Peletier, Mark Small volume-fraction limit of the diblock copolymer problem: II. Diffuse-interface functional. SIAM J. Math. Anal. 43 (2011), no. 2, 739-763.
[5] Choksi, Rustum; Ren, Xiaofeng On the derivation of a density functional theory for microphase separation of diblock copolymers. J. Statist. Phys. 113 (2003), no. 1-2, 151-176.
[6] Choksi, Rustum; Ren, Xiaofeng Diblock copolymer/homopolymer blends: derivation of a density functional theory. Phys. D 203 (2005), no. 1-2, 100-119.
[7] Cicalese, Marco; Spadaro, Emanuele N.; Zeppieri Caterina I. Asymptotic analysis of a second-order singular perturbation model for phase transitions Calc. Var. 41 (2011), 127-150.
[8] Fonseca, Irene; Tartar, Luc The gradient theory of phase transitions for systems with two potential wells. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), no. 1-2, 89-102.
[9] Giuliani, Alessandro; Lebowitz, Joel L.; Lieb, Elliott H. Periodic minimizers in 1D local mean field theory. Comm. Math. Phys. 286 (2009), no. 1, 163-177.
[10] Gurtin, Morton Some results and conjectures in the gradient theory of phase transitions
[11] Leizarowitz, Arie Infinite horizon autonomous systems with unbounded cost. Appl. Math. Optim. 19 (1985), no. 1, 19-43.
[12] Leizarowitz, Arie; Mizel, Victor J. One-dimensional infinite-horizon variational problems arising in continuum mechanics. Arch. Rational Mech. Anal. 106 (1989), no. 2, 161-194.
[13] Leoni, Giovanni Lecture notes on Γ-convergence and liquid-liquid phase transitions. CNA-PIRE Working Group Meetings (2012).
[14] Mizel, V. J.; Peletier, L. A.; Troy, W. C. Periodic phases in second-order materials. Arch. Ration. Mech. Anal. 145 (1998), no. 4, 343-382.
[15] Modica, Luciano The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98 (1987), no. 2, 123-142.
[16] Modica, Luciano; Mortola, Stefano Un esempio di Γ-convergenza. (Italian) Boll. Un. Mat. Ital. B (5) 14 (1977), no. 1, 285-299. journal
[17] Müller, Stefan Singular perturbations as a selection criterion for periodic minimizing sequences. Calc. Var. Partial Differential Equations 1 (1993), no. 2, 169-204
[18] Peletier, L. A.; Troy, W. C. Spatial patterns described by the extended Fisher-Kolmogorov equation: periodic solutions. SIAM J. Math. Anal. 28 (1997), no. 6, 1317-1353.
[19] Ren, Xiaofeng; Wei, Juncheng On the multiplicity of solutions of two nonlocal variational problems. SIAM J. Math. Anal. 31 (2000), no. 4, 909-924
[20] Ren, Xiaofeng; Wei, Juncheng On energy minimizers of the diblock copolymer problem. Interfaces Free Bound. 5 (2003), no. 2, 193-238
[21] Sternberg, Peter The effect of a singular perturbation on nonconvex variational problems. Arch. Rational Mech. Anal. 101 (1988), no. 3, 209-260.
[22] Tartar, Luc An Introduction to the Homogenization Method in Optimal Design Optimal Shape Design, Lecture Notes in Maths. 1740, A. Cellina & A. Ornelas eds, 47-156, Springer, 2000.
[23] Tartar, Luc On Homogenization and Γ-convergence Homogenization 2001, Proceedings of the First HMS2000 International School and Conference on Homogenization. Naples, Complesso Monte S. Angelo, June 18-22 and 23-27, 2001
[24] Tartar, Luc Unpublished note 1987
[25] van der Waals, J.D. Translation of J. D. van der Waals' "The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density''. J. Statist. Phys. 20 (1979)
Variational methods for phase transitions and copolymers
Tuesdays, 2:30pm - 3:30pm
Wean Hall 7218
We will investigate the application of variational methods to the study and understanding of phase transitions in materials and phase separation in copolymers.
Resources
[1] Alberti, Giovanni; Choksi, Rustum; Otto, Felix Uniform energy distribution for an isoperimetric problem with long-range interactions. J. Amer. Math. Soc. 22 (2009), no. 2, 569-605.
[2] Chermisi, M.; Dal Maso, G.; Fonseca, I.; Leoni, G. Singuar perturbation models in phase transitions for second order materials (2010).
[3] Choksi, Rustum; Peletier, Mark Small volume fraction limit of the diblock copolymer problem: I. Sharp-interface functional. SIAM J. Math. Anal. 42 (2010), no. 3, 1334-1370.
[4] Choksi, Rustum; Peletier, Mark Small volume-fraction limit of the diblock copolymer problem: II. Diffuse-interface functional. SIAM J. Math. Anal. 43 (2011), no. 2, 739-763.
[5] Choksi, Rustum; Ren, Xiaofeng On the derivation of a density functional theory for microphase separation of diblock copolymers. J. Statist. Phys. 113 (2003), no. 1-2, 151-176.
[6] Choksi, Rustum; Ren, Xiaofeng Diblock copolymer/homopolymer blends: derivation of a density functional theory. Phys. D 203 (2005), no. 1-2, 100-119.
[7] Cicalese, Marco; Spadaro, Emanuele N.; Zeppieri Caterina I. Asymptotic analysis of a second-order singular perturbation model for phase transitions Calc. Var. 41 (2011), 127-150.
[8] Fonseca, Irene; Tartar, Luc The gradient theory of phase transitions for systems with two potential wells. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), no. 1-2, 89-102.
[9] Giuliani, Alessandro; Lebowitz, Joel L.; Lieb, Elliott H. Periodic minimizers in 1D local mean field theory. Comm. Math. Phys. 286 (2009), no. 1, 163-177.
[10] Gurtin, Morton Some results and conjectures in the gradient theory of phase transitions
[11] Leizarowitz, Arie Infinite horizon autonomous systems with unbounded cost. Appl. Math. Optim. 19 (1985), no. 1, 19-43.
[12] Leizarowitz, Arie; Mizel, Victor J. One-dimensional infinite-horizon variational problems arising in continuum mechanics. Arch. Rational Mech. Anal. 106 (1989), no. 2, 161-194.
[13] Leoni, Giovanni Lecture notes on Γ-convergence and liquid-liquid phase transitions. CNA-PIRE Working Group Meetings (2012).
[14] Mizel, V. J.; Peletier, L. A.; Troy, W. C. Periodic phases in second-order materials. Arch. Ration. Mech. Anal. 145 (1998), no. 4, 343-382.
[15] Modica, Luciano The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98 (1987), no. 2, 123-142.
[16] Modica, Luciano; Mortola, Stefano Un esempio di Γ-convergenza. (Italian) Boll. Un. Mat. Ital. B (5) 14 (1977), no. 1, 285-299. journal
[17] Müller, Stefan Singular perturbations as a selection criterion for periodic minimizing sequences. Calc. Var. Partial Differential Equations 1 (1993), no. 2, 169-204
[18] Peletier, L. A.; Troy, W. C. Spatial patterns described by the extended Fisher-Kolmogorov equation: periodic solutions. SIAM J. Math. Anal. 28 (1997), no. 6, 1317-1353.
[19] Ren, Xiaofeng; Wei, Juncheng On the multiplicity of solutions of two nonlocal variational problems. SIAM J. Math. Anal. 31 (2000), no. 4, 909-924
[20] Ren, Xiaofeng; Wei, Juncheng On energy minimizers of the diblock copolymer problem. Interfaces Free Bound. 5 (2003), no. 2, 193-238
[21] Sternberg, Peter The effect of a singular perturbation on nonconvex variational problems. Arch. Rational Mech. Anal. 101 (1988), no. 3, 209-260.
[22] Tartar, Luc An Introduction to the Homogenization Method in Optimal Design Optimal Shape Design, Lecture Notes in Maths. 1740, A. Cellina & A. Ornelas eds, 47-156, Springer, 2000.
[23] Tartar, Luc On Homogenization and Γ-convergence Homogenization 2001, Proceedings of the First HMS2000 International School and Conference on Homogenization. Naples, Complesso Monte S. Angelo, June 18-22 and 23-27, 2001
[24] Tartar, Luc Unpublished note 1987
[25] van der Waals, J.D. Translation of J. D. van der Waals' "The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density''. J. Statist. Phys. 20 (1979)