Research

Vortex lattice

Recent prize: 2026 AMS Fellow

CV:cv25

CVshort:

Research interests:

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.

Most recently, with my collaborators, Stan Alama and Silas Vriend, we have introduced a new type of partitioning problems where clusters have more than one “infinite” phase. This problem comes from the study of block co-polymer in the dilute regime, and has led to new exciting development in the study of clusters.

 PUBLICATIONS:

Submitted preprints in peer-reviewed journals

Minimizing solutions of degenerate Allen-Cahn equations with three wells in R^2, Lia Bronsard, Etienne Sandier and Peter Sternberg, (2025), arXiv:2509.08111

Compensation effects for anisotropic energies of two-dimensional unit vector fields, Lia Bronsard, Dmitry Golovaty, Xavier Lamy and Peter Sternberg, (2025), arXiv:2507.17345

On the non-uniqueness of locally minimizing clusters via singular cones,  Lia Bronsard, Robin Neumayer, Michael Novack and Anna Skorobogatova, (2025), arXiv:2507.13995

Decorated phases in triblock copolymers: zeroth- and first-order analysis,  Stanley Alama, Lia Bronsard, Xinyang Lu and Chong Wang, (2025), arXiv:2503.21684

Interaction energies in nematic liquid crystal suspensions,  Lia Bronsard, Xavier Lamy, Dominik Stantejsky* and Raghavendra Venkatraman, (2025), arXiv:2501.14043

Minimizing Harmonic Maps on the Unit Ball with Tangential Anchoring,  Lia Bronsard, Andrew Colinet* and Dominik Stantejsky*, (2025). arXiv:2501.11565

Asymptotics for Minimizers of Landau-de Gennes with Magnetic Field and Tangential Anchoring,  Lia Bronsard, Dean Louizos* and Dominik Stantejsky*,
(2024), arXiv:2410.09914,  accepted for publication in SIAM J. Math. Anal.

Structure of Saturn ring defects for two small spherical colloidal particles,  Lia Bronsard, Spencer Locke*, Hayley Monson* and Dominik Stantejsky*,  accepted in C.R.Acad. Sci., (2024)

• “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. Article
• “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. Article
• “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. Article
• “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. Article
• “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. Article