Research

My papers, loosely organized by topic, are below. You can also find them on the arXiv, where I try to keep things up to date. I've been lucky to work with great collaborators, including: Matthew Badger, Simon Bortz, Otis Chodosh, Guy David, Guido De Philippis, Nick Edelen, Xavier Fernández-Real, Max Goering, Cole Jeznach, Aapo Kauranen, Svitlana Mayboroda, Dana Mendelson, Robin Neumayer, Martí Prats, Georgios Sakellaris, Yannick Sire, Mariana Smit Vega Garcia, Luca Spolaor, Tatiana Toro, Bozhidar Velichkov, Hui Yu, Zihui Zhao.

Harmonic Measure and Analysis on Rough Sets

• M. Badger, M. Engelstein, T. Toro "Slowly vanishing mean oscillations: non-uniqueness of blow-ups in a two-phase free boundary problem." To appear in the Vietnam Journal of Mathematics for their Special Issue in honor of Carlos Kenig’s 70th Birthday. (2023)

• M. Engelstein, C. Jeznach, S. Mayboroda "Non-local distance functions and geometric regularity." Submitted. (2022)

•  S. Bortz, M. Engelstein M. Goering, T. Toro and Z. Zhao "Two Phase Free Boundary Problem for Poisson Kernels." Indiana University Math J. (2022)

• G. David, M. Engelstein, S. Mayboroda "Square functions, non-tangential limits and harmonic measure in co-dimensions larger than one" Duke Math J. (2021).

•  M. Badger, M. Engelstein, T. Toro, "Regularity of the singular set in a two-phase problem for harmonic measure with Hölder data" Revisita Matematica Ib. (2020). 

• S. Bortz, M. Engelstein, "Reifenberg Flatness and Oscillation of the unit Normal Vector" Not For Publication. Superseded by BEGTZ above. (2017).

•  M. Engelstein, "Parabolic NTA Domains in R2" Communications in PDE. (2017).

• M. Engelstein "A Free Boundary Problem for the Parabolic Poisson Kernel" Advances in Math. (2017). 

• M. Badger, M. Engelstein, T. Toro  "Structure of Sets which are Well Approximated by Zero Sets of Harmonic Polynomials" Analysis & PDE. (2017). 

• M. Engelstein, "A Two-Phase Free Boundary Problem For Harmonic Measure" Ann. Sci. de l'ENS. (2016).

Variational Free Boundary Problems

• M. Engelstein, X. Fernández-Real, H. Yu "Graphical solutions to one-phase free boundary problems". Crelle's Journal (2023)

• G. De Philippis, M. Engelstein, L. Spolaor, B. Velichkov "Rectifiability and almost everywhere uniqueness of the blow-up for the vectorial Bernoulli free boundaries" Submitted to Nonlinear Analysis for their Special Issue on Free Boundary Problems (2021)

• G. David, M. Engelstein, M. Smit Vega Garcia, T. Toro, “Branch Points for (Almost-)Minimizers of Two-Phase Free Boundary Problems" Forum of Math: Sigma (2023)

• G. David, M. Engelstein M. Smit Vega Garcia and T. Toro, “Regularity for almost-minimizers of variable coefficient Bernoulli-type functionals." Math Zeit. (2021).

• M. Engelstein A. Kauranen, M. Prats, G. Sakellaris and Y. Sire. “Minimizers for the thin one-phase free boundary problem." CPAM. (2021).

• M. Engelstein, L. Spolaor and B. Velichkov. “Uniqueness of the blow-up at isolated singularities for the Alt-Caffarelli Functional." Duke Math J. (2020). Oberwolfach Report

• G. David, M. Engelstein, T. Toro. “Free Boundary Regularity for Almost-Minimizers." Adv. Math. (2019).

• N. Edelen, M. Engelstein “Quantitative stratification for some free-boundary problems." Trans. A.M.S. (2019)

Misc Geometric PDE

• O. Chodosh, M. Engelstein, L. Spolaor “The Riemannian Quantitative Isoperimetric Inequality." JEMS. (2023)

• M. Engelstein, R. Neumayer, L. Spolaor “Quantitative Stability for Minimizing Yamabe Metrics." Trans. A.M.S. (2022)

• M. Engelstein, D. Mendelson, “Non-uniqueness of bubbling for wave maps." Ars Inveniendi Analytica (2022)

• M. Engelstein, L. Spolaor, B. Velichkov, “(Log-)Epiperimetric Inequality and Regularity over Smooth Cones For Almost Area-Minimizing Currents" Geometry & Topology (2019).

Pre-Graduate School Publications

• M. Engelstein, “The Least-Perimeter Partition of a Sphere into Four Equal Areas." Discrete Comput. Geom. (2010) 

• Q. Maurmann, M. Engelstein, A. Marcuccio, T. Pritchard “Asymptotics of Perimeter-Minimizing Partitions." Can. Math. Bull. (2010).

• M. Engelstein, Q. Maurmann, A. Marcuccio, T. Pritchard “Isoperimetric problems on the sphere and on surfaces with density." N.Y.J.M. (2009)

Support: Currently supported by the NSF DMS CAREER 2143719. I was previously supported by an NSF Standard Grant (DMS 2000288), an NSF Postdoc (DMS-1703306), an NSF GRFP and a DoD NDSEG fellowship.