Broadly speaking my work lies in the intersection of Geometric Measure Theory, Geometric Analysis, Harmonic Analysis, and PDEs.
I work on regularity problems of elliptic PDEs and generalized minimal submanifolds. In particular I am interested in the regularity of area-minimizing currents; unique continuation problems; the properties of the harmonic and elliptic measures and their relations to elliptic boundary value problem.
A complete list of my preprints can also be viewed on my arXiv page.
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Expansion of Harmonic Functions Near the Boundary of Dini Domains
with Carlos Kenig
Submitted on arXiv
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Elliptic Measures for Dahlberg-Kenig-Pipher Operators: Asymptotically Optimal Estimates
with Simon Bortz, Tatiana Toro
Submitted on arXiv
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Boundary Unique Continuation on Dini Domains and the Size of the Singular Set
with Carlos Kenig
Submitted on arXiv
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Optimal Poisson Kernel Regularity for Elliptic Operators with Hölder-continuous Coefficients in Vanishing Chord-arc Domains
with Simon Bortz, Tatiana Toro
Submitted on arXiv
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Uniform Rectifiability and Elliptic Operators Satisfying a Carleson Measure Condition. Part II: The Large Constant Case
with Steve Hofmann, José María Martell, Svitlana Mayboroda, Tatiana Toro
(Combined with Part I of the series for publication) Geom. Funct. Anal. (2021) 31:325-401
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Two Phase Free Boundary Problem for Poisson Kernels
with Simon Bortz, Max Engelstein, Max Goering, Tatiana Toro
To appear on Indiana U. Math. J.
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Dirichlet Energy-minimizers with Analytic Boundary
with Camillo De Lellis
Submitted on arXiv
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Dirichlet Problem in Domains with Lower Dimensional Boundaries
with Joseph Feneuil, Svitlana Mayboroda
Rev. Mat. Iberoam. (2021) 37:821-910
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Square Function Estimates, BMO Dirichlet Problem, and Absolute Continuity of Harmonic Measure on Lower-dimensional Sets
with Svitlana Mayboroda
Anal. PDE (2019) 12:1843-1890
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Uniform Rectifiability and Elliptic Operators Satisfying a Carleson Measure Condition. Part I: The Small Constant Case
with Steve Hofmann, José María Martell, Svitlana Mayboroda, Tatiana Toro
Submitted on arXiv
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Boundary Rectifiability and Elliptic Operators with W^{1,1} Coefficients
with Tatiana Toro
Adv. Calc. Var. (2021) 14:37-62
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BMO Solvability and the A-infty Condition of the Elliptic Measure in Uniform Domains
J. Geom. Anal. (2018) 28:866-908