The geometry of mesoscopic phase transition interfaces

Previous Article
On the intersection of homoclinic classes on singular-hyperbolic sets
DCDS Home
This Issue
Next Article
Rates of convergence towards the boundary of a self-similar set

December 2007, 19(4): 777-798. doi: 10.3934/dcds.2007.19.777

The geometry of mesoscopic phase transition interfaces

Matteo Novaga ^1, and Enrico Valdinoci ^2,

1.	Dipartimento di Matematica Università di Pisa, Via Buonarroti 2, I-56127 Pisa, Italy
2.	Dipartimento di Matematica, Università di Roma "Tor Vergata", Via della Ricerca Scientifica 1, I-00133 Roma, Italy

Received November 2006 Revised March 2007 Published September 2007

Abstract
Related Papers

We consider a mesoscopic model of phase transitions and investigate the geometric properties of the interfaces of the associated minimal solutions. We provide density estimates for level sets and, in the periodic setting, we construct minimal interfaces at a universal distance from any given hyperplane.

Keywords: plane-like solutions., Ginzburg-Landau-Allen-Cahn equation, density estimates.

Mathematics Subject Classification: Primary: 35B10, 35B15, 35B27; Secondary: 35J20, 35Q72, 49S0.

Citation: Matteo Novaga, Enrico Valdinoci. The geometry of mesoscopic phase transition interfaces. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 777-798. doi: 10.3934/dcds.2007.19.777

[1]	Hirokazu Ninomiya. Entire solutions and traveling wave solutions of the Allen-Cahn-Nagumo equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2001-2019. doi: 10.3934/dcds.2019084
[2]	Antonin Chambolle, Gilles Thouroude. Homogenization of interfacial energies and construction of plane-like minimizers in periodic media through a cell problem. Networks & Heterogeneous Media, 2009, 4 (1) : 127-152. doi: 10.3934/nhm.2009.4.127
[3]	Isabeau Birindelli, Enrico Valdinoci. On the Allen-Cahn equation in the Grushin plane: A monotone entire solution that is not one-dimensional. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 823-838. doi: 10.3934/dcds.2011.29.823
[4]	Hans G. Kaper, Peter Takáč. Bifurcating vortex solutions of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 871-880. doi: 10.3934/dcds.1999.5.871
[5]	Sen-Zhong Huang, Peter Takáč. Global smooth solutions of the complex Ginzburg-Landau equation and their dynamical properties. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 825-848. doi: 10.3934/dcds.1999.5.825
[6]	Christos Sourdis. On the growth of the energy of entire solutions to the vector Allen-Cahn equation. Communications on Pure & Applied Analysis, 2015, 14 (2) : 577-584. doi: 10.3934/cpaa.2015.14.577
[7]	Paul H. Rabinowitz, Ed Stredulinsky. On a class of infinite transition solutions for an Allen-Cahn model equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 319-332. doi: 10.3934/dcds.2008.21.319
[8]	Eleonora Cinti. Saddle-shaped solutions for the fractional Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 441-463. doi: 10.3934/dcdss.2018024
[9]	Gianni Gilardi. On an Allen-Cahn type integrodifferential equation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 703-709. doi: 10.3934/dcdss.2013.6.703
[10]	Shujuan Lü, Chunbiao Gan, Baohua Wang, Linning Qian, Meisheng Li. Traveling wave solutions and its stability for 3D Ginzburg-Landau type equation. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 507-527. doi: 10.3934/dcdsb.2011.16.507
[11]	Hongzi Cong, Jianjun Liu, Xiaoping Yuan. Quasi-periodic solutions for complex Ginzburg-Landau equation of nonlinearity $\|u\|^{2p}u$. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 579-600. doi: 10.3934/dcdss.2010.3.579
[12]	N. I. Karachalios, Hector E. Nistazakis, Athanasios N. Yannacopoulos. Asymptotic behavior of solutions of complex discrete evolution equations: The discrete Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 711-736. doi: 10.3934/dcds.2007.19.711
[13]	Giorgio Fusco. Layered solutions to the vector Allen-Cahn equation in $\mathbb{R}^2$. Minimizers and heteroclinic connections. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1807-1841. doi: 10.3934/cpaa.2017088
[14]	Giorgio Fusco, Francesco Leonetti, Cristina Pignotti. On the asymptotic behavior of symmetric solutions of the Allen-Cahn equation in unbounded domains in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 725-742. doi: 10.3934/dcds.2017030
[15]	Michał Kowalczyk, Yong Liu, Frank Pacard. Towards classification of multiple-end solutions to the Allen-Cahn equation in $\mathbb{R}^2$. Networks & Heterogeneous Media, 2012, 7 (4) : 837-855. doi: 10.3934/nhm.2012.7.837
[16]	Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127
[17]	N. Maaroufi. Topological entropy by unit length for the Ginzburg-Landau equation on the line. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 647-662. doi: 10.3934/dcds.2014.34.647
[18]	Jingna Li, Li Xia. The Fractional Ginzburg-Landau equation with distributional initial data. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2173-2187. doi: 10.3934/cpaa.2013.12.2173
[19]	Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 665-682. doi: 10.3934/cpaa.2005.4.665
[20]	Jun Yang. Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2359-2388. doi: 10.3934/dcds.2014.34.2359

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences