Publications

2026

Elliptic theory and dimension drop for operators with divergence-free form-bounded drifts in one-sided chord-arc domains (with Alberto Pacati). submitted
The Dirichlet problem with boundary data of fractional smoothness in domains with Ahlfors regular boundary (with Ariel Barton and Alberto Pacati). submitted
Scalar Dissipation Criticality in Compressible Magnetized Turbulence (with David N. Spergel and Camillo De Lellis). submitted
Physics-Informed Neural Embeddings of PDE Solution Families (with Raul Jimenez, Pavlos Protopapas, Leonid Sarieddine, David N. Spergel, Pedro Tarancón-Álvarez). submitted
Level sets of fractional Sobolev functions (with Camillo De Lellis and Ming-Yuan Chang). submitted
What is…Robin harmonic measure? (with Max Engelstein and Marcel Filoche). Notices of the American Mathematical Society. accepted
Regularity thresholds for anomalous dissipation and related phenomena in passive scalars (with Marco Bagnara, Daniel W. Boutros, Camillo De Lellis). submitted
Random divergence-free drifts and the Onsager-Richardson threshold (with Daniel W. Boutros and Camillo De Lellis). submitted
Using wavelet decomposition to determine the dimension of structures (with David Spergel). PNAS, (2026).

2025

Periodic homogenization and harmonic measures (with Guy David, Antoine Gloria, Siguang Qi). J. Geom. Anal.
Robin harmonic measure with a variable permeability parameter (with Alberto Pacati). accepted
Robin Green Function Estimates and a model of Mammalian Lungs (with Guy David, Stefano Decio, Max Engelstein, Marcel Filoche and Marco Michetti). submitted
The Poisson-Dirichlet problem in domains with Ahlfors regular boundary (with Ariel Barton and Alberto Pacati). The Proceedings of the 9th European Congress of Mathematics.
The landscape function on ℝ^d (with Guy David and Antoine Gloria). Les Annales de la Faculté des Sciences de Toulouse, (2025).

2024

Dimension and structure of the Robin Harmonic Measure on Rough Domains (with Guy David, Stefano Decio, Max Engelstein, Marco Michetti). submitted
Non-local distance functions and geometric regularity (with Max Engelstein and Cole Jeznach). Advances in Mathematics, (2024).
Critical Perturbations for Second Order Elliptic Operators. Part II: Non-tangential maximal function estimates (with Simon Bortz, Steve Hofmann, José Luis Luna Garcia, Bruno Poggi). Archive for Rational Mechanics and Analysis, (2024).
The Anderson mobility edge as a percolation transition (Marcel Filoche, Pierre Pelletier, Dominique Delande, Svitlana Mayboroda). Physical Review B, (2024).

2023

Small A_∞ results for Dahlberg-Kenig-Pipher operators in sets with uniformly rectifiable boundaries (with Guy David and Linhan Li). Trans. Am. Math. Soc., (2023).
The Regularity problem in domains with lower dimensional boundaries (with Zanbing Dai and Joseph Feneuil). Journal of Functional Analysis, (2023).
Green functions and smooth distances (with Joseph Feneuil and Linhan Li). Math. Ann., (2023).
Transference of scale-invariant estimates from Lipschitz to Non-tangentially accessible to Uniformly rectifiable domains (with Steve Hofmann and Jose Maria Martell). Anal. PDE, (2023).
On an Effective Equation of the Reduced Hartree-Fock Theory (with Ilias Chenn, Wei Wang, and Shiwen Zhang). Adv. Nonlinear Stud., Special Issue in honor of David Jerison, (2023).
Elliptic theory in domains with boundaries of mixed dimension (with Guy David and Joseph Feneuil). Asterisque, (2023).
Carleson Perturbations for the Regularity problem (with Zanbing Dai and Joseph Feneuil). Revista Matematica Iberoamericana, (2023).

2022

The landscape law for tight binding Hamiltonians (with Douglas Arnold, Marcel Filoche, Wei Wang, Shiwen Zhang). Communications in Mathematical Physics, (2022).
Carleson estimates for the Green function on domains with lower dimensional boundaries (with Guy David and Linhan Li). Journal of Functional Analysis, 283 (5), 109553 (2022).
Spectral functions and localization landscape theory in speckle potentials (with P. Pelletier, D. Delande, V. Josse, A. Aspect, D. Arnold, M. Filoche). Phys Rev A, 105, 023314 (2022).
Approximation of Green functions and domains with uniformly rectifiable boundaries of all dimensions (with Guy David). Advances in Math, 410 (2022).
Harmonic measure is absolutely continuous with respect to the Hausdorff measure on all low-dimensional uniformly rectifiable sets (with Guy David). International Mathematics Research Notices, (2022).
Wigner-Weyl description of light absorption in disordered semiconductor alloys using the localization landscape theory (with Jean-Philippe Banon, Pierre Pelletier, Claude Weisbuch, Marcel Filoche). Phys. Rev. B, 105, 125422 (2022).
Green function estimates on complements of low-dimensional uniformly rectifiable sets (with Guy David and Joseph Feneuil). Math. Ann. (2022).
L^p theory for the square roots and square functions of elliptic operators having a BMO anti-symmetric part (with Steve Hofmann, Linhan Li, and Jill Pipher). Math. Z. 301, 935–976 (2022).

2021

Carleson measure estimates for the Green function (with Guy David and Linhan Li). Archive for Rational Mechanics and Analysis,243, 1525–1563.
Critical Perturbations for Second Order Elliptic Operators. I: Square function bounds for layer potentials (with Simon Bortz, Steve Hofmann, Jose Luis Luna Garcia, and Bruno Poggi). Analysis & PDE.
The landscape law for the integrated density of states (with Guy David and Marcel Filoche). Advances in Mathematics, 390, (2021).
Sharp estimates for the integrated density of states in Anderson tight-binding models (with P. Desforges, S. Zhang, G. David, D. Arnold, W. Wang, M. Filoche). Phys. Rev. A, 104, 012207 (2021).
The Dirichlet problem for elliptic operators having a BMO anti-symmetric part (with Steve Hofmann, Linhan Li, and Jill Pipher). Math. Ann., (2021).
Uniform rectifiability and elliptic operators satisfying a Carleson measure condition (with Steve Hofmann, Jose Maria Martell, Tatiana Toro, and Zihui Zhao). Geom. Funct. Anal. 31, 325–401 (2021).
Good elliptic operators on Cantor sets (with Guy David). Advances in Mathematics, 383, (2021).
The effective potential of an M-matrix (with Marcel Filoche and Terence Tao). Journal of Mathematical Physics, 62, 041902 (2021).
Carleson perturbations of elliptic operators on domains with low dimensional boundaries (with Bruno Poggi). Journal of Functional Analysis, 280, Issue 8 (2021).
Dirichlet problem in domains with lower dimensional boundaries (with Joseph Feneuil and Zihui Zhao). Revista Matematica Iberoamericana, 37, 3, 821–910 (2021).
Square functions, non-tangential limits and harmonic measure in co-dimensions larger than one (with Guy David and Max Engelstein). Duke Math. J., 170 (3), 455–501 (2021).
Elliptic theory for sets with higher co-dimensional boundaries (with Guy David and Joseph Feneuil). Memoirs of the AMS.

2020

Reply to comment by Comtet and Texier on “The effective confining potential of quantum states in disordered media” (with Doug Arnold, Guy David, Marcel Filoche, and David Jerison). Physical Review Letters, 124, 219802 (2020).
Nontangential estimates on layer potentials and the Neumann problem for higher order elliptic equations (with Ariel Barton and Steve Hofmann). International Mathematics Research Notices, (2020).

2019

Uniform rectifiability and elliptic operators satisfying a Carleson measure condition. Part I: The small constant case (with Steve Hofmann, Jose Maria Martell, Tatiana Toro, Zihui Zhao).submitted
Universality of fold-encoded localized vibrations in enzymes (with Yann Chalopin, Francesco Piazza, Claude Weisbuch, and Marcel Filoche). Scientific Reports, Volume 9, Number 1 (2019).
Bounds on layer potentials with rough inputs for higher order elliptic equations (with Ariel Barton and Steve Hofmann). Proceedings of the London Mathematical Society, 3, 119, 613–653 (2019).
Dahlberg’s theorem in higher co-dimension (with Guy David and Joseph Feneuil). Journal of Functional Analysis, 276, Number 9 (2019).
Exponential decay estimates for fundamental solutions of Schrödinger-type operators (with Bruno Poggi). Transactions of the AMS, Volume 372, Number 6 (2019).
Dirichlet and Neumann boundary values of solutions to higher order elliptic equations (with Ariel Barton and Steve Hofmann). Annales de l’Institut Fourier, 69, Number 4 (2019).
Square function estimates, BMO Dirichlet problem, and absolute continuity of harmonic measure on lower-dimensional sets (with Zihui Zhao). Analysis & PDE, 12, Number 7 (2019).
A new elliptic measure on lower dimensional sets (with Guy David and Joseph Feneuil). Acta Mathematica Sinica, special issue in honor of the 65th birthday of Professor Carlos Kenig, 35 (2019).
Localization of eigenfunctions via an effective potential (with Douglas Arnold, Guy David, Marcel Filoche, David Jerison). Communications in Partial Differential Equations, 44, Number 11 (2019).
Computing spectra without solving eigenvalue problems (with Douglas Arnold, Guy David, Marcel Filoche, David Jerison). SIAM Journal on Scientific Computing, 41, Number 1 (2019).

2018

The effect of disorder and irregularities on solutions to boundary value problems and spectra of differential operators. Proceedings of the ICM – Rio de Janeiro 2018, Vol. III, Invited lectures, 1691–1712 (2018).
Polyharmonic capacity and Wiener test of higher order (with Vladimir Maz’ya). Inventiones Mathematicae, (2018) 211: 779.
Fundamental Matrices and Green Matrices for non-homogeneous elliptic systems (with Blair Davey and Jonathan Hill). Publicacions Matematiques, Volume 62, Number 2 (2018), 537–614.
The Neumann problem for higher order elliptic equations with symmetric coefficients (with Ariel Barton and Steve Hofmann). Math. Ann., (2018) 371: 297.

2017

Localization landscape theory of disorder in semiconductors III: Application to carrier transport and recombination in light emitting diodes (with Chi-Kang Li, Marco Piccardo, Li-Shuo Lu, Lucio Martinelli, Jacques Peretti, James S. Speck, Claude Weisbuch, Marcel Filoche, Yuh-Renn Wu). Phys. Rev. B, 95, 144206 (2017).
Localization landscape theory of disorder in semiconductors I: Theory and modeling (with Marcel Filoche, Marco Piccardo, Yuh-Renn Wu, Chi-Kang Li, Claude Weisbuch). Phys. Rev. B, 95, 144204 (2017).
A free boundary problem for the localization of eigenfunctions (with Guy David, Marcel Filoche, David Jerison). Asterisque, 392 (2017), 203 pages.
Square function estimates on layer potentials for higher-order elliptic equations (with Ariel Barton and Steve Hofmann). Mathematische Nachrichten, 290: 2459–2511.
Local Hardy spaces associated with inhomogeneous higher order elliptic operators (with Dachun Yang and Jun Cao). Anal. Appl. (Singap.) 15 (2017), no. 2, 137–224.
Localization of eigenfunctions. Notices of the AMS, (2017).

2016

Absolute continuity between the surface measure and harmonic measure implies rectifiability (with Steve Hofmann, Jose Maria Martell, Xavier Tolsa and Alexander Volberg).submitted
One single static measurement predicts wave localization in complex structures (with Gautier Lefebvre, Alexane Gondel, Marc Dubois, Michael Atlan, Florian Feppon, Aimé Labbé, Camille Gillot, Alix Garelli, Maxence Ernoult, Marcel Filoche, Patrick Sebbah). Physical Review Letters, 117, 074301 (2016).
Higher-order elliptic equations in non-smooth domains: some old and new results (with Ariel Barton). Harmonic Analysis, PDE, Complex Analysis, Banach Spaces, and Operator Theory. Celebrating Cora Sadosky’s life. Volume 1.
Rectifiability of harmonic measure (with Jonas Azzam, Steve Hofmann, Jose Maria Martell, Mihalis Mourgoglou, Xavier Tolsa, and Alexander Volberg). Geom. Funct. Anal. 26 (2016), no. 3, 703–728.
Harmonic measure is rectifiable if it is absolutely continuous with respect to the co-dimension one Hausdorff measure (with Jonas Azzam, Steve Hofmann, Jose Maria Martell, Mihalis Mourgoglou, Xavier Tolsa, and Alexander Volberg). C. R. Math. Acad. Sci. Paris 354 (2016), no. 4, 351–355.
The effective confining potential of quantum states in disordered media (with Doug Arnold, Guy David, Marcel Filoche, and David Jerison). Physical Review Letters, 116 (2016), 056602.
Layer potentials and boundary-value problems for second order elliptic operators with data in Besov spaces (with Ariel Barton). Mem. Amer. Math. Soc. 243 (2016), no. 1149, v+110 pp.
Uniform Rectifiability, Carleson measure estimates, and approximation of harmonic functions (with Steve Hofmann and Jose Maria Martell). Duke Math. J. 165 (2016), no. 12, 2331–2389.
Maximal function characterizations of Hardy spaces associated to homogeneous higher order elliptic operators (with Dachun Yang and Jun Cao). Forum Math. 28 (2016), no. 5, 823–856.

2015

Layer potentials and boundary value problems for elliptic equations with complex bounded coefficients satisfying the small Carleson measure norm condition (with Steve Hofmann and Mihalis Mourgoglou). Advances in Mathematics, 480–564.
Dual hidden landscapes for Anderson localization in discrete lattices (with Marcel Filoche, Marcelo Lyra). Europhys. Lett. EPL, Volume 109, Number 4, Editor’s choice.
The regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients (with Steve Hofmann, Carlos Kenig, and Jill Pipher). Mathematische Annalen, 361 (2015), no. 3‑4, 863–907.
Square function / non-tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators (with Steve Hofmann, Carlos Kenig, and Jill Pipher). Journal of the American Mathematical Society, 28 (2015), no. 2, 483–529.

2014

Regularity of solutions to the polyharmonic equation in general domains (with Vladimir Maz’ya). Inventiones Mathematicae, 196 (2014), no. 1, 1–68.
Uniform Rectifiability and Harmonic Measure III: Riesz transform bounds imply uniform rectifiability of boundaries of 1-sided NTA domains (with Steve Hofmann and Jose Maria Martell). International Mathematics Research Notices, 2014, no. 10, 2702–2729.
Boundary-value problems for higher-order elliptic equations in non-smooth domains (with Ariel Barton). Oper. Theory Adv. Appl., 236, Birkhäuser Springer, Basel, 2014.

2013 and Earlier

The Dirichlet problem for higher order equations in composition form (with Ariel Barton). Journal of Functional Analysis, Volume 265, Issue 1, (2013), 49–107.
The landscape of Anderson localization in a disordered medium (with Marcel Filoche). Contemporary Mathematics, 601 (2013), 113–121.
Universal mechanism for Anderson and weak localization (with Marcel Filoche). Proceedings of the National Academy of Sciences, (2012), 109 (37) 14761–14766.
Localization of eigenfunctions of a one-dimensional elliptic operator (with Marcel Filoche and Brandon Patterson). Contemporary Mathematics, 581 (2012), 99–116.
Second order elliptic operators with complex bounded measurable coefficients in L^p, Sobolev and Hardy spaces (with Steve Hofmann and Alan McIntosh). Les Annales Scientifiques de l’École Normale Supérieure, Volume 44, fascicule 5 (2011), 723–800.
The connections between Dirichlet, Regularity and Neumann problems for second order elliptic operators with complex bounded measurable coefficients. Advances in Mathematics, 225 (2010), 1786–1819.
Boundedness of the gradient of a solution and Wiener test of order one for the biharmonic equation (with Vladimir Maz’ya). Inventiones Mathematicae, 175 (2009), no. 2, 287–334.
Strong localization induced by one clamped point in thin plate vibrations (with Marcel Filoche). Physical Review Letters, Volume 103, Issue 25, Article 254301, (2009).
Finite square function implies integer dimension (with Alexander Volberg). C. R. Math. Acad. Sci. Paris, 347 (2009), pp. 1271–1276.
Hardy and BMO spaces associated to divergence form elliptic operators (with Steve Hofmann). Mathematische Annalen, 344 (2009), no. 1, 37–116.
Boundedness of the square function and rectifiability (with Alexander Volberg). C. R. Math. Acad. Sci. Paris, 347 (2009), pp. 1051–1056.
Pointwise estimates for the polyharmonic Green function in general domains (with Vladimir Maz’ya). Operator Theory: Advances and Applications 193, 143–158 (2009).
Boundedness of the Hessian of a biharmonic function in a convex domain (with Vladimir Maz’ya). Comm. Partial Differential Equations 33 (2008), no. 7‑9, 1439–1454.
Interpolation of Hardy-Sobolev-Besov-Triebel-Lizorkin Spaces and applications to problems in partial differential equations (with Nigel Kalton and Marius Mitrea). Contemp. Math., 2007.
The Poisson problem for the Lamé system on low-dimensional Lipschitz domains (with Marius Mitrea). Integral methods in science and engineering, 137–160, Birkhauser Boston, 2006.
Layer potentials and boundary value problems for Laplacian in Lipschitz domains with data in quasi-Banach Besov spaces (with Marius Mitrea). Annali di Matematica Pura ed Applicata (4) 185 (2006), no. 2, 155–187.
Sharp estimates for Green potentials on non-smooth domains (with Marius Mitrea). Mathematical Research Letters, 11 (2004), 481–492.
Square-function estimates for singular integrals and applications to partial differential equations (with Marius Mitrea). Differential Integral Equations, 17 (2004), no. 7‑8, 873–892.
On one approach to the solution of problems of numerical analysis of the electrostatic field. Collection of the scientific works of KISP, V. 6 (2001), 223–227.