Quantitative stratification for some free-boundary problems
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- by Nick Edelen and Max Engelstein PDF
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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
- H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105–144. MR 618549
- Hans Wilhelm Alt, Luis A. Caffarelli, and Avner Friedman, Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc. 282 (1984), no. 2, 431–461. MR 732100, DOI 10.1090/S0002-9947-1984-0732100-6
- Jonas Azzam and Xavier Tolsa, Characterization of $n$-rectifiability in terms of Jones’ square function: Part II, Geom. Funct. Anal. 25 (2015), no. 5, 1371–1412. MR 3426057, DOI 10.1007/s00039-015-0334-7
- Luis A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. I. Lipschitz free boundaries are $C^{1,\alpha }$, Rev. Mat. Iberoamericana 3 (1987), no. 2, 139–162. MR 990856, DOI 10.4171/RMI/47
- Luis A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. II. Flat free boundaries are Lipschitz, Comm. Pure Appl. Math. 42 (1989), no. 1, 55–78. MR 973745, DOI 10.1002/cpa.3160420105
- Luis A. Caffarelli, David Jerison, and Carlos E. Kenig, Global energy minimizers for free boundary problems and full regularity in three dimensions, Noncompact problems at the intersection of geometry, analysis, and topology, Contemp. Math., vol. 350, Amer. Math. Soc., Providence, RI, 2004, pp. 83–97. MR 2082392, DOI 10.1090/conm/350/06339
- Luis A. Caffarelli and Carlos E. Kenig, Gradient estimates for variable coefficient parabolic equations and singular perturbation problems, Amer. J. Math. 120 (1998), no. 2, 391–439. MR 1613650
- Jeff Cheeger and Aaron Naber, Lower bounds on Ricci curvature and quantitative behavior of singular sets, Invent. Math. 191 (2013), no. 2, 321–339. MR 3010378, DOI 10.1007/s00222-012-0394-3
- L. Caffarelli, H. Shahgholian, and K. Yeressian, A minimization problem with free boundary related to a cooperative system, https://arxiv.org/abs/1608.07689, 2016.
- G. David, M. Engelstein, and T. Toro, Free boundary regularity for almost-minimizers, 2017. arXiv:1702.06580.
- C. de Lellis, A. Marchese, E. Spadaro, and D. Valtorta, Rectifiability and upper Minkowski bounds for singularities of harmonic q-valued maps, 2016. arXiv:1612.01813.
- Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, Mathematical Surveys and Monographs, vol. 38, American Mathematical Society, Providence, RI, 1993. MR 1251061, DOI 10.1090/surv/038
- Guy David and Tatiana Toro, Reifenberg parameterizations for sets with holes, Mem. Amer. Math. Soc. 215 (2012), no. 1012, vi+102. MR 2907827, DOI 10.1090/S0065-9266-2011-00629-5
- G. David and T. Toro, Regularity of almost minimizers with free boundary, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 455–524. MR 3385167, DOI 10.1007/s00526-014-0792-z
- N. Edelen, A. Naber, and D. Valtorta, Quantitative Reifenberg theorem for measures, arXiv:1612.08052.
- Peter W. Jones, Rectifiable sets and the traveling salesman problem, Invent. Math. 102 (1990), no. 1, 1–15. MR 1069238, DOI 10.1007/BF01233418
- David Jerison and Ovidiu Savin, Some remarks on stability of cones for the one-phase free boundary problem, Geom. Funct. Anal. 25 (2015), no. 4, 1240–1257. MR 3385632, DOI 10.1007/s00039-015-0335-6
- Dennis Kriventsov and Fanghua Lin, Regularity for shape optimizers: The nondegenerate case, Preprint, 09 2016.
- D. Kinderlehrer and L. Nirenberg, Regularity in free boundary problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), no. 2, 373–391. MR 440187
- M. Miskiewicz, Discrete Reifenberg-type theorem, 2016. arXiv:1612.02461.
- Dario Mazzoleni, Susanna Terracini, and Bozhidar Velichkov, Regularity of the optimal sets for some spectral functionals, Geom. Funct. Anal. 27 (2017), no. 2, 373–426. MR 3626615, DOI 10.1007/s00039-017-0402-2
- A. Naber and D. Valtorta, The singular structure and regularity of stationary and minimizing varifolds, 2015. arXiv:1505.03428.
- A. Naber and D. Valtorta, Stratification for the singular set of approximate harmonic maps, 2016. arXiv:1611.03008.
- Aaron Naber and Daniele Valtorta, Rectifiable-Reifenberg and the regularity of stationary and minimizing harmonic maps, Ann. of Math. (2) 185 (2017), no. 1, 131–227. MR 3583353, DOI 10.4007/annals.2017.185.1.3
- Daniela De Silva and David Jerison, A singular energy minimizing free boundary, J. Reine Angew. Math. 635 (2009), 1–21. MR 2572253, DOI 10.1515/CRELLE.2009.074
- Georg Sebastian Weiss, Partial regularity for a minimum problem with free boundary, J. Geom. Anal. 9 (1999), no. 2, 317–326. MR 1759450, DOI 10.1007/BF02921941
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