In the field of Partial Differential Equations and the Calculus of Variations, we are interested in studying qualitative properties of solution of equations arising in various contexts: physics, geometry, biology, or economics, for instance. In the Calculus of Variations, the problem is to minimize a quantity (called a functional,) such as arclength of curves on surfaces, and use analysis and ODE/PDE techniques to find the qualitative property of the minima. We can also study the dynamics of systems which decrease the values of the functional in time, and approach the optimal solutions asymptotically. For the example of arclength, this dynamical process is called motion by curvature. In this way, analysis, physics, and geometry enter into the study of these equations. Another examples comes from the study of vortices, which are point singularities with quantized winding numbers, occuring in the Ginzburg-Landau model of superconductors. Again, the optimal shape of vortex solutions is a minimization problem in the Calculus of Variations, and there are associated equations governing the motion of vortices.
The analysis of the structure and evolution of point or line singularities in partial differential equations (PDE) problems has led to a wealth of new mathematical tools, results and questions over the past several years. One important class of PDEs arises from Ginzburg–Landau models: the state of a physical system is described by an “order parameter” (a vector-valued function) and a free energy functional with a small parameter epsilon.
I am interested in gradient flows associated to this energy and its stationary equilibria, in the limit as epsilon tends to zero, especially the qualitative information on the singularities (vortices or interfaces separating regions of different phase.) Often these problems lead to challenging analytical questions as well as nice geometrical results for the singularities thus obtained.
Accepted/submitted preprints in peer-reviewed journals
- Spherical particle in a nematic liquid crystal under an external field: the Saturn ring regime, Stan Alama, Lia Bronsard, Xavier Lamy. arXiv:1710.04756, preprint 2017. Submitted to Journal of Nonlinear Sciences.
- Droplet breakup in the liquid drop model with background potential, Stan Alama, Lia Bronsard, Rustum Choksi, Ihsan Topaloglu, arXiv:1708.04292, preprint 2017. Submitted to Calculus of Variations and PDE.
- Droplet phase in a nonlocal isoperimetric problem under confinement, S. Alama, L. Bronsard, R. Choksi, I. Topaloglu, arXiv:1609.03589, preprint 2016. Submitted to Indiana Univ Math J.
- Ground-states for the liquid drop and TFDW models with long-range attraction, Stan Alama, Lia Bronsard, Rustum Choksi, Ihsan Topaloglu, arXiv:1707.06674, Journal of Math Physics. Volume 58, Issue 10, (2017): 103503.
- Sharp Interface Limit of an Energy Modelling Nanoparticle-Polymer Blends”}, S. Alama, L. Bronsard, I. Topaloglu, arXiv:1508.01206, Interfaces and Free Boundaries.
Volume 18, Issue 2, (2016), pp. 263–290, DOI: 10.4171/IFB/364
- “Minimizers of the Landau-de Gennes energy around a spherical colloid particle”, S. Alama, L. Bronsard, X. Lamy, preprint 2015, arXiv:1504.00421, Arch Rat Mech Anal. 222 (2016), no. 1, 427–450.
“A Degenerate Isoperimetric Problem and Traveling Waves to a Bi-stable Hamiltonian System”, S. Alama, L. Bronsard, A. Contreras, J. Dadok, P. Sternberg, arXiv:1504.00423, accepted in Comm on Pure and Applied Math.
- “Vortex structure in p-wave superconductors”, S. Alama, L. Bronsard, X. Lamy, preprint 2014, arxiv.org: 1411.3665v1, J. Math. Phys.} 56 (2015), no. 11, 111503, 20 pp.
- Weak Anchoring for a Two-Dimensional Liquid Crystal”, S. Alama, L. Bronsard, B. Galvao-Sousa, Accepted for publication in Nonlinear Analysis: T.M.A. August 2014.ABGS_Qtensor
Published articles in peer-reviewed journals
- “Domain walls in the coupled Gross-Pitaevskii equations”, S. Alama, L. Bronsard, A. Contreras, D. Pelinovsky, Arch. Ration. Mech. Anal., 215 (2015), no. 2, 579-610.205_2014_789
- “Singular Limits for Thin Film Superconductors in Strong Magnetic Fields. ” S. Alama, L. Bronsard, B. Galvo-Sousa, Asymptotic Analysis, 83, no. 1-2, (2013), 127-156.1209.3696v1
- ”On compound vortices in a two-component Ginzburg–Landau functiona.l” S. Alama, L. Bronsard, P. Mironescu, Indiana Univ. Math. Journal, 61, No. 5 (2012), 18611909. ABM
- ”Minimizers of the Lawrence-Doniach Functional with Oblique Magnetic Fields.” S. Alama, L. Bronsard and E. Sandier, Comm. Math. Phys. (2012) (DOI) 10.1007/s00220- 011-1399-2.
- ”On the Lawrence-Doniach Model of Superconductivity: Magnetic Fields Parallel to the Axes.” S. Alama, L. Bronsard, E. Sandier, Jour. of the European Math. Soc (JEMS) 14 (2012), no. 6, 18251857. ABS
- ”Gamma-convergence of 2D Ginzburg-Landau functionals with vortex concentration along curves.” S. Alama, L. Bronsard, V. Millot, Journal d’Analyse Mathématique, 114 (2011), 341391 ABM
- ”Thin film limits for Ginzburg–Landau with strong applied magnetic fields.” S. Alama, L. Bronsard, B. Galvo-Sousa, SIAM Jour. of Mathematical Analysis, Vol. 42 (2010), No. 1, pp. 97124.
- “Periodic Minimizers of the Anisotropic Ginzburg-Landau Model”, S. Alama, L. Bron- sard and E. Sandier, Calc. Var. Partial Differential Equations 36 (2009), no. 3, 399–417.
- “ On the structure of fractional degree vortices in a spinor Ginzburg-Landau model,” S. Alama, L. Bronsard, P. Mironescu, Journal of Functional Analysis 256 (2009), pp. 1118-1136.
- “Vortices for a rotating toroidal Bose–Einstein Condensate,” S. Alama, L. Bronsard, A.J. Montero, Arch. Rat. Mech. Anal., vol. 187 (2008), no. 3, pp. 481-522.
- “ On the shape of interlayer vortices in the Lawrence-Doniach model,” with S. Alama and E. Sandier. Trans. Amer. Math. Soc. 360 (2008), no. 1, 1–34 (electronic).
- “Fractional degree vortices for a spinor Ginzburg-Landau model,” with S. Alama. Commun. Contemp. Math. 8 (2006), no. 3, 355–380.
- “On the Ginzburg-Landau model of a superconducting ball in a uniform field,” with S. Alama and J. Alberto Montero. Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), no. 2, 237–267.
- “Vortices and pinning effects for the Ginzburg-Landau model in multiply connected domains,” with S. Alama. Comm. Pure Appl. Math. 59 (2006), no. 1, 36–70.
- “ Giant vortex and the breakdown of strong pinning in a rotating Bose-Einstein con- densate,” with A. Aftalion and S. Alama. Arch. Ration. Mech. Anal. 178 (2005), no. 2, 247–286.
- “ Pinning effects and their breakdown for a Ginzburg-Landau model with normal inclusions”, with S. Alama. J. Math. Phys. 46 (2005), no. 9, 095102, 39 pp.
- “Vortices and the lower critical field for a Ginzburg–Landau model with ferromagnetic interactions,” with S. Alama. Proc. Roy. Soc. Edinburgh Sect. A, vol. 135 (2005), no. 2, pp. 223-252.
- “Long-time behavior for Competition-Diffusion systems via viscosity comparison”, with S.-A Shim, preprint, 2003. Discrete Contin. Dyn. Syst. 13 (2005), no. 3, 561–581.
- “On the second critical field for a Ginzburg–Landau model with ferromagnetic interactions,” with S. Alama. Rev. Math. Phys., vol. 16, No. 2 (2004), 147-174.
- “Des vortex fractionnaires pour un modele Ginzburg–Landau spineur / Half-integer Vortices for a Spin-coupled Ginzburg–Landau model,” with S. Alama. C. R. Acad. Sci. Paris, Serie 1, vol. 337 (2003), 243–247.
- “Minimizers of the Lawrence–Doniach energy in the small-coupling limit: finite width samples in a parallel field”, with S. Alama and J. Berlinsky. Annales IHP-Analyse nonlineaire, vol. 19 (2002), 281–312.
- “Symmetric Vortex solutions in the U(1) and SO(5) Ginzburg–Landau Models of Superconductivity,” with S. Alama. In Nonlinear PDE’s in Condensed Matter and Reactive Flows, H. Berestycki et Y. Pomeau (eds.),pp. 323–337, Kluwer Academic Publishers, 2002.
- “Periodic vortex lattices for the Lawrence–Doniach model of layered superconductors in
a parallel field”, with S. Alama and J. Berlinsky. Commun. Contemp. Math., vol. 3 (2001), no. 3, 457–494.
- “Vortices with antiferromagnetic cores in the $SO(5)$ theory of superconductivity”, with S. Alama, J. Berlinsky, and T. Giorgi. Phys. Rev. B. vol. 60, no. 9, pp. 6901–6906, 1999.
- “Analysis of some macroscopic models of high–$T_c$ superconductivity,” with S. Alama.CRM Proceedings and Lecture Notes, AMS, vol. 27, pp.1–16, 2001.
- “Vortex Structures for an SO(5) Model of High-T_C Superconductivity and Antiferromagnetism”, with S. Alama and T. Giorgi. Proc. Roy. Soc. Edin., ser. A, vol. 130 (2000), no. 6, 1183–1215.
- “Uniqueness of Symmetric Vortex Solutions in the Ginzburg–Landau Model of Superconductivity,” with S. Alama and T. Giorgi. Journal of Functional Analysis, vol. 167, pp. 399–424, 1999.
- “A multi-phase Mullins-Sekerka system: matched asymptotic expansions and an implicit time discretization for the geometric evolution problem”, with H. Garcke and B. Stoth, Proc. of the Royal Soc. of Edinborough, Vol 128A, pp. 481–506, 1998.
- “The Singular Limit of a Vector-Valued Reaction-Diffusion Process”, with B. Stoth, Trans. AMS , Vol 350, no. 12, pp. 4931–4953, 1998.
- “A Singular Limit of the Ginzburg-Landau Equations for Superconductivity and the one-phase Stefan problem”, with B. Stoth, Annales IHP-Analyse nonlin\’eaire, Vol 15,no. 3, pp. 371–397, 1998.
- Slow motion in the gradient theory of phase transitions via energy and spectrum,”, with N. Alikakos and G. Fusco, Calc. of Variation and PDE, Vol 6, pp. 39–66, 1998.
- “Volume Preserving Mean Curvature Flow as a Limit of a Nonlocal Ginzburg-Landau Equation”, with B. Stoth, SIAM J. Math. Anal., Vol 28, no 4, pp.769-807, July 1997.
- “Stationary layered solutions in $\bold R^2$ for an Allen-Cahn system with multiple well potential” with S. Alama and C. Gui, Calc. of Variation and PDE, vol. 5, pp 359-390, 1997.
- “A Three Layered Minimizer in $\bold R^2$ for a Variational Problem with a Symmetric Three Well Potential”, with C. Gui and M. Schatzman, Comm. Pure Appld. Math., vol 49, pp 677-715, 1996.
- “On the Existence of High Multiplicity Interfaces”, with B. Stoth, Math. Res. Lett., vol 3, pp 41-50, 1996.
- “A Numerical Method for Tracking Curve Networks Moving with Curvature Motion”, with B. Wetton, Jour. Comp. Phys., vol 120, pp 66-87, 1995.
- “On Three-Phase Boundary Motion and the Singular Limit of a Vector-Valued Ginzburg-Landau Equation,” with F. Reitich, Arch Rat. Mech. and Analysis, vol 124, no 4, pp 355-379, 1993.
- “Front Propagation for Reaction-Diffusion Equations of Bistable Type”, with G. Barles and P. E. Souganidis, Ann. I. H. P.-non lineaire, vol 9 no 5, pp 479-496, 1992.
- “On the Slow Dynamics for the Cahn-Hilliard Equation in One Space Dimension”, with D. Hilhorst, Proc. Roy. Soc. Lon.-series A (math. phys. Sci.}, vol 439 no 1907, pp 669-682, 1992.
- “Motion by Mean Curvature as the Singular Limit of Ginzburg-Landau Dynamics”, with R. V. Kohn, Jour. of Diff. Eq., vol. 90, pp. 211-237, 1991.
- “On the Slowness of Phase Boundary Motion in One Space Dimension”, with R. V. Kohn,Comm. Pure Appl. Math., vol. XLIII, pp. 983-997, 1990.