Research

My research interests lie at the interface of Analysis, Partial Differential Equations (PDE), and Geometric Measure Theory (GMT).  I am interested in understanding the exact relations between the geometry of sets, the structure of the underlying operators, and the behavior of solutions.

You can find my publications on arxiv.

Here is my Google Scholar page. 

Preprints

• The $L^p$ Poisson-Neumann problem and its relation to the Neumann problem (with Joseph Feneuil).  arXiv:2406.16735

Publications

•  Boundary behavior of solutions of elliptic operators in divergence form with a BMO anti-symmetric part  (with Jill Pipher).Comm. Partial Differential Equations, 44, no. 2 (2019): 156-204.  arXiv:1712.06705

• $\ell^2$ decoupling in R^2 for curves with vanishing curvature (with Chandan Biswas, Maxim Gilula, Jeremy Schwend, and Yakun Xi). Proc. Amer. Math. Soc. 148, no. 5 (2020): 1987-1997. arXiv:1812.04760

•  The Dirichlet problem for elliptic operators having a BMO anti-symmetric part, (with S. Hofmann, S. Mayboroda, J. Pipher). Math. Ann. (2021) DOI: 10.1007/s00208-021-02219-1.  arXiv:1908.08587

• $L^p$ theory for the square roots and square functions of elliptic operators having a BMO anti-symmetric part (with S. Hofmann, S. Mayboroda, J. Pipher).    Math. Z. 301, pages 935–976 (2022). DOI:10.1007/s00209-021-02938-w    arXiv:1908.01030

• Carleson measure estimates for the Green function, (with G. David and S. Mayboroda). Arch. Ration. Mech. Anal. 243, pages1525–1563 (2022).  DOI: 10.1007/s00205-021-01746-0. arXiv:2102.09592

•  Carleson estimates for the Green function on domains with lower dimensional boundaries,  (with G. David and S. Mayboroda). J. Funct. Anal. Volume 283, Issue 5 (2022). DOI: 10.1016/j.jfa.2022.109553  arXiv:2107.08101

• Small $A_\infty$ results for weak Dahlberg-Kenig-Pipher operators in sets with uniformly rectifiable boundaries, (with G. David, S. Mayboroda).  Trans. Amer. Math. Soc. (2023) DOI: 10.1090/tran/9004  arxiv:2207.13602

• Green functions and smooth distances, (with J. Feneuil, S. Mayboroda).  Math. Ann. 389 (2024), no.3, 2637–2727. DOI: 10.1007/s00208-023-02715-6 arXiv:2211.05318 

• A Green function characterization of uniformly rectifiable sets of any codimension, (with J. Feneuil). Adv. Math.  430, (2023) Paper No. 109220, DOI: 10.1016/j.aim.2023.109220 arXiv:2302.14087, slides

Other articles

My PhD Thesis: $L^p$ Dirichlet problem for second order elliptic operators having a BMO anti-symmetric part.