Research

Research Interests

My research interests are in partial differential equations, harmonic analysis, and geometric measure theory. Much of my work is related to the (quantitative) unique continuation properties of solutions to elliptic and parabolic equations, but I am interested in the broader applications of Carleman estimates as well. I also work on the solvability of elliptic boundary value problems, the connections between elliptic and parabolic theory. More recently, I have been studying fractal sets and generalizations of the Besicovitch projection theorem.

My work is supported in part by the National Science Foundation CAREER grant, DMS-2236491 and National Science Foundation LEAPS-MPS grant, DMS-2137743.

Papers

20. Self-similar sets and Lipschitz graphs.

with Silvia Ghinassi and Bobby Wilson

19. Variable-coefficient parabolic theory as a high-dimensional limit of elliptic theory

with Mariana Smit Vega Garcia

(To appear in Calculus of Variations and Partial Differential Equations.)

18. On Landis' conjecture in the plane for potentials with growth

(To appear in Vietnam Journal of Mathematics.)

17. Exponential Decay Estimates for Fundamental Matrices of Generalized Schrödinger Systems

with Joshua Isralowitz

(To appear in Mathematische Annalen.)

16. Improved quantitative unique continuation for complex-valued drift equations in the plane

with Carlos Kenig and Jenn-Nan Wang

(Forum Math. 34 (2022), no. 6, 1641–1661.)

15. Upper and lower bounds on the rate of decay of the Favard curve length for the four-corner Cantor set

with Laura Cladek and Krystal Taylor

(Indiana Univ. Math. J. 71 (2022), no. 3, 1003–1025.)

14. A Quantification of a Besicovitch Nonlinear Projection Theorem via Multiscale Analysis 

with Krystal Taylor

(J. Geom. Anal. 32 (2022), no. 4, 138. )

13. Strong unique continuation for the Lamé system with less regular coefficients

with Ching-Lung Lin and Jenn-Nan Wang

(Math. Ann. 381 (2021), no. 1-2, 1005–1029. .)

12. On Landis' conjecture in the plane for some equations with sign-changing potentials

(Rev. Mat. Iberoam. 36 (2020), no. 5, 1571–1596.)

11. Quantitative unique continuation for Schrödinger operators

(Journal of Functional Analysis 279 (2020), no. 4, 108566. )

10. Landis' conjecture for general second order elliptic equations with singular lower order terms in the plane

with Jenn-Nan Wang

(Journal of Differential Equations 268 (2020), no. 3, 977-1042.)

9. Quantitative uniqueness of solutions to second order elliptic equations with singular lower order terms

with Jiuyi Zhu

(Comm. Partial Differential Equations, 44 (2019), no. 11, 1217–1251. )

8. On Landis' conjecture in the plane when the potential has an exponentially decaying negative part

with Carlos Kenig and Jenn-Nan Wang

(Algebra i Analiz 31 (2019), no. 2, 204–226.)

7. Liouville-type theorem for the Lamé system with singular coefficients

with Ching-Lung Lin and Jenn-Nan Wang

(Proceedings of the American Mathematical Society, 147 (2019), no. 6, 2619–2624.)

6. Quantitative uniqueness of solutions to second order elliptic equations with singular potentials in two dimensions

with Jiuyi Zhu

(Calculus of Variations and Partial Differential Equations. 57 (2018), no. 3, 57:92.)

5. Fundamental matrices and Green matrices for non-homogeneous elliptic systems

with Jonathan Hill and Svitlana Mayboroda

(Publicacions Matemàtiques, 62 (2018), 537–614.)

4. Parabolic theory as a high-dimensional limit of elliptic theory 

(Archive for Rational Mechanics and Analysis, 228 (2018), no. 1, 159–196. 35 (53))

3. The Landis conjecture for variable coefficient second-order elliptic PDEs

with Carlos Kenig and Jenn-Nan Wang 

(Transactions of the American Mathematical Society, 369(11):8209-8237, 2017.)

2. A Meshkov-type construction for the borderline case 

(Differential and Integral Equations, 28 (2015), no. 3-4, 271-290.)

1. Some quantitative unique continuation results for eigenfunctions of the magnetic Schrödinger operator 

(Communications in Partial Differential Equations, 39 (2014), no. 5, 876-945.)