## Quantitative stratification for some free-boundary problems

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- by Nick Edelen and Max Engelstein PDF
- Trans. Amer. Math. Soc.
**371**(2019), 2043-2072 Request permission

## Abstract:

In this paper we prove the rectifiability of and measure bounds on the singular set of the free-boundary for minimizers of a functional first considered by Alt–Caffarelli [J. Reine Angew. Math. 325 (1981), pp. 105–144]. Our main tools are the Quantitative Stratification and Rectifiable-Reifenberg framework of Naber–Valtorta [Ann. of Math. (2) 185 (2017), pp. 131–227], which allow us to do a type of “effective dimension-reduction”. The arguments are sufficiently robust that they apply to a broad class of related free-boundary problems as well.## References

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## Additional Information

**Nick Edelen**- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139-4307
- MR Author ID: 1099014
- Email: nedelen@mit.edu
**Max Engelstein**- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139-4307
- MR Author ID: 868968
- Email: maxe@mit.edu
- Received by editor(s): February 24, 2017
- Received by editor(s) in revised form: September 12, 2017
- Published electronically: October 26, 2018
- Additional Notes: The first author was supported by NSF grant DMS-1606492. The second author was partially supported by NSF Grant No. DMS-1440140 while the author was in residence at MSRI in Berkeley, California, during Spring 2017.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**371**(2019), 2043-2072 - MSC (2010): Primary 35R35
- DOI: https://doi.org/10.1090/tran/7401
- MathSciNet review: 3894044