Research

Research interests:

  • Nonlinear partial differential equations
  • Calculus of variations
  • Applied mathematics

I am interested in applying comprehensive techniques from nonlinear PDEs and geometric measure theory to understand complex singularity structures in certain physical systems, including superconductivity, liquid crystals, thin film blisters, convection pattern formations and hyperbolic conservation laws. The studies of these systems are highly interdisciplinary. The mathematical studies of such problems require the development of new mathematical tools, and these studies further foster the fundamental understanding in related fields of sciences. My work currently evolves along three lines:

  • The first line of my research is centered around a second order singular perturbation problem, called the Aviles-Giga functional, in connection with the theory of smectic liquid crystals and thin film blisters. A similar energy functional, called the Regularized Cross-Newell energy, occurs in convection pattern formations. I am interested in the variational analysis of these problems, leading to better understanding of the singularity structures in the physical systems.
  • The second line of my focus explores the connections between certain notions of convexity, including polyconvexity, quasiconvexity and rank-one convexity, and regularity properties of nonlinear PDEs that can be formulated as differential inclusions. The geometric aspects carried by these various notions of convexity have fascinating connections to uniqueness and regularity of certain nonlinear PDEs, including certain hyperbolic conservation laws.
  • The third line of my research concerns three-dimensional superconductivity models. My work has been focusing on an anisotropic model called the Lawrence-Doniach model, which models highly anisotropic superconductors with layered structure. I am also interested in the variational analysis to understand the singularity structures in these superconductors. However the techniques are completely different from those used in the analysis of the Aviles-Giga functional due to the different nature of singularities in these systems.

Publications and preprints:

1. Rigidity of a non-elliptic differential inclusion related to the Aviles-Giga conjecture. (with X. Lamy and A. Lorent) Submitted.

https://arxiv.org/abs/1910.10284

2. On the Rank-1 convex hull of a set arising from a hyperbolic system of Lagrangian elasticity. (with A. Lorent) Submitted.

https://arxiv.org/abs/1909.05938

3. First critical field of highly anisotropic three-dimensional superconductors via a vortex density model. (with A. Contreras) SIAM J. Math. Anal., 51 (2019), no. 6, 4490--4519.

https://doi.org/10.1137/19M1237521

4. Null Lagrangian measures in subspaces, compensated compactness and conservation laws. (with A. Lorent) Arch. Ration. Mech. Anal., 234 (2019), no. 2, 857--910.

https://doi.org/10.1007/s00205-019-01403-7

5. Analysis of minimizers of the Lawrence-Doniach energy for superconductors in applied fields. (with P. Bauman) Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), no. 11, 5903--5926.

https://doi.org/10.3934/dcdsb.2019112

6. Bounded solutions for a class of Hamiltonian systems. (with P. Korman) Electron. J. Qual. Theory Differ. Equ. (2018), no. 81, 1--7.

https://doi.org/10.14232/ejqtde.2018.1.81

7. Regularity of the Eikonal equation with two vanishing entropies. (with A. Lorent) Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 2, 481--516.

https://doi.org/10.1016/j.anihpc.2017.06.002

8. Convergence of the Lawrence-Doniach energy for layered superconductors with magnetic fields near H_{c1} . SIAM J. Math. Anal. 49 (2017), no. 2, 1225--1266.

https://doi.org/10.1137/16M1064398

9. Rigidity of the Eikonal equation with two L^p entropies. (with X. Lamy and A. Lorent) Preprint.

10. Analysis of superconductivity models in magnetic fields. Ph.D. thesis, Purdue University, August 2014.

Invited talks:

  • Rigidity of the Eikonal equation subject to two $L^p$ entropies, Minisymposium on Singular Solutions to Geometric Problems in Continuum and Discrete Mechanics, SIAM Conference on Analysis of PDEs, December 2019 (scheduled).
  • Null Lagrangian measures in subspaces with applications to a system of conservation laws, 82nd Midwest PDE Seminar, Purdue University, October 2018.
  • On the Aviles-Giga functional and regularity of its zero energy states, Colloquium, New Mexico State University, August 2018.
  • Regularity of the Eikonal equation with two vanishing entropies, Minisymposium on Variational Problems from Materials Science, SIAM Conference on Mathematical Aspects of Materials Science, July 2018.
  • On the first critical field of a 3D anisotropic superconductivity model, Special Session on Singularities and Phase transitions in Nonlinear PDE's, 2018 CMS Summer Meeting, June 2018.
  • Gamma-convergence for an anisotropic superconductivity model with magnetic fields near $H_{c_1}$, Minisymposium on Modeling and Analysis of Condensed Matter Systems, 2017 SIAM Conference on Analysis of PDEs, December 2017.
  • Regularity of the Eikonal equation with two vanishing entropies, Special Session on Analysis of Variational Problems and Nonlinear Partial Differential Equations, AMS Spring Central Sectional Meeting, Indiana University, April 2017.
  • Regularity of the Eikonal equation with two vanishing entropies, PDE-Analysis Seminar, University of Kentucky, November 2016.
  • Regularity of the Eikonal equation with two vanishing entropies, Special Session on Nonlinear PDEs in Material Science and Mathematical Biology, AMS Fall Eastern Sectional Meeting, Bowdoin College, September 2016.
  • Analysis of minimizers of the Lawrence-Doniach model for layered superconductors in magnetic fields, Minisymposium on the Ginzburg-Landau Theory and Related Topics, 2015 SIAM Conference on Analysis of PDEs, December 2015.
  • Analysis of the Lawrence-Doniach energy for layered superconductors in magnetic fields, Minisymposium on the Ginzburg-Landau Model and Related Topics, The 8th International Congress on Industrial and Applied Mathematics, August 2015.
  • A lower bound for the Lawrence-Doniach energy with magnetic fields in the lower regime, PDE Seminar, Purdue University, April 2015.
  • Analysis of the Lawrence-Doniach model for layered superconductors in magnetic fields, Special Session on Calculus of Variations, Nonlinear PDEs, and Applications, AMS Spring Central Sectional Meeting, Michigan State University, March 2015.
  • Properties of minimizers of the Lawrence-Doniach energy in perpendicular magnetic fields, Harmonic Analysis and Differential Equations Seminar, University of Illinois, October 2013.