Abstract
We survey some recent results on variational and evolution problems concerning a certain class of convex 1-homogeneous functionals for vector-valued maps related to models in phase transitions (Hele-Shaw), superconductivity (Ginzburg-Landau) and superfluidity (Gross-Ktaevskii). Minimizers and gradient flows of such functionals may be characterized as solutions of suitable non-local vectorial generalizations of the classical obstacle problem.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
G. Aronsson, Minimization problem for the functional supx F (x, f (x), f ′(x)), Ark. Mat. 6 (1965), 33–53.
P. Athavale, R. L. Jerrard, M. Novaga and G. Orlandi, Weighted TV minimization and applications to vortex density models, preprint, 2013.
A. Aftalion and R. L. Jerrard, Shape of vortices for a rotating Bose-Einstein condensate, Physical Review A (2) 66 (2002), 023611/1-023611/7.
F. Alter, V. Caselles and A. Chambolle, A characterization of convex calibrables sets in ℝ N, Math. Annalen 332 (2005), 329–366.
S. Baldo, R. L. Jerrard, G. Orlandi and H. M. Soner, Convergence of Ginzburg-Landau functionals in 3-d superconductivity, Archive Rat. Mech. Analysis (3) 205 (2012), 699–752.
S. Baldo, R. L. Jerrard, G. Orlandi and H. M. Soner, Vortex density models for superconductivity and superfluidity, Comm. Math. Phys. (1) 318 (2013), 131–171.
E. N. Barron, L. C. Evans and R. Jensen, The infinity Laplacian, Aronsson’s equation and their generalizations, Trans. Amer. Math. Soc. (1) 360 (2008), 77–101.
A. Briani, A. Chambolle, M. Novaga and G. Orlandi, On the gradient flow of a one-homogeneous functional, Confluentes Mathematici (4) 3, 617–635.
H. Brézis, “Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert”, North-Holland, Amsterdam, 1973.
G. Carlier and M. Comte, On a weighted total variation minimization problem, J. Funct. Anal. (1) 250 (2007), 214–226.
V. Caselles, A. Chambolle and M. Novaga, Total Variation in imaging, In: “Handbook of Mathematical Methods in Imaging”, Springer, 2011, 1016–1057.
V. Caselles, G. Facciolo and E. Meinhardt, Anisotropic Cheeger Sets and Applications, SLAM J. Imaging Sciences, (4) 2 (2009), 1211–1254.
D. Chiron, Boundary problems for the Ginzburg-Landau equation, Commun. Contemp. Math. 7 (2005), 597–648.
C. M. Elliott and V. Janovský, A variational inequality approach to Hele-Shaw flow with a moving boundary, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 93–107.
I. Ionescu and T. Lachand-Robert, Generalized Cheeger sets related to landslides, Calc. Var. Partial Differential Equations (2) 23 (2005), 227–249.
B. Gustafsson, Applications ofVariational inequalities to a moving boundary problem for Hele Shaw flows, Siam J. Math. Anal. 16 (1985), 279–300.
C. I. Kim and A. Mellet, Homogenization of a Hele-Shaw problem in periodic and random media, Arch. Rat. Mech. Anal. 194 (2009), 507–530.
Y. Meyer, “Oscillating Patterns in Image Processing and Nonlinear Evolution Equations”, University Lecture Series, 22. American Mathematical Society, Providence, RI, 2001.
A. Montero and B. Stephens, On the geometry of Gross-Pitaevskii vortex curves for generic data, Proceedigs of the A.M.S., to appear, 2012.
Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. (1) 22 (2009), 167–210.
L. Rudin, S. J. Osher and E. FATEMI, Nonlinear total variation based noise removal algorithms, Physica D 60 (1992), 259–268.
S. K. Smirnov, Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows, Rossiĭskaya Akademiya Nauk. Algebra i Analiz 5 (1993), 206–238.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Scuola Normale Superiore Pisa
About this paper
Cite this paper
Novaga, M., Orlandi, G. (2013). Limiting models in condensed matter Physics and gradient flows of 1-homogeneous functional. In: Chambolle, A., Novaga, M., Valdinoci, E. (eds) Geometric Partial Differential Equations proceedings. CRM Series, vol 15. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-473-1_11
Download citation
DOI: https://doi.org/10.1007/978-88-7642-473-1_11
Publisher Name: Edizioni della Normale, Pisa
Print ISBN: 978-88-7642-472-4
Online ISBN: 978-88-7642-473-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)