Boundary structure and size in terms of interior and exterior harmonic measures in higher dimensions
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- by C. Kenig, D. Preiss and T. Toro;
- J. Amer. Math. Soc. 22 (2009), 771-796
- DOI: https://doi.org/10.1090/S0894-0347-08-00601-2
- Published electronically: April 25, 2008
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Abstract:
In this work we introduce the use of powerful tools from geometric measure theory (GMT) to study problems related to the size and structure of sets of mutual absolute continuity for the harmonic measure $\omega ^+$ of a domain $\Omega =\Omega ^+\subset \mathbb {R}^n$ and the harmonic measure $\omega ^-$ of $\Omega ^-$, $\Omega ^-=\mbox {int}(\Omega ^c)$, in dimension $n\ge 3$.References
- 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 BKPW. Beckner, C. Kenig & J. Pipher, unpublished manuscript.
- Christopher J. Bishop, Some questions concerning harmonic measure, Partial differential equations with minimal smoothness and applications (Chicago, IL, 1990) IMA Vol. Math. Appl., vol. 42, Springer, New York, 1992, pp. 89–97. MR 1155854, DOI 10.1007/978-1-4612-2898-1_{7}
- C. J. Bishop, L. Carleson, J. B. Garnett, and P. W. Jones, Harmonic measures supported on curves, Pacific J. Math. 138 (1989), no. 2, 233–236. MR 996199
- John E. Brothers and William P. Ziemer, Minimal rearrangements of Sobolev functions, J. Reine Angew. Math. 384 (1988), 153–179. MR 929981
- Lennart Carleson, On the support of harmonic measure for sets of Cantor type, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 113–123. MR 802473, DOI 10.5186/aasfm.1985.1014
- Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
- Kenneth Falconer, Techniques in fractal geometry, John Wiley & Sons, Ltd., Chichester, 1997. MR 1449135
- John B. Garnett and Donald E. Marshall, Harmonic measure, New Mathematical Monographs, vol. 2, Cambridge University Press, Cambridge, 2005. MR 2150803, DOI 10.1017/CBO9780511546617
- Robert Hardt and Leon Simon, Nodal sets for solutions of elliptic equations, J. Differential Geom. 30 (1989), no. 2, 505–522. MR 1010169
- Peter W. Jones, A geometric localization theorem, Adv. in Math. 46 (1982), no. 1, 71–79. MR 676987, DOI 10.1016/0001-8708(82)90054-8
- Peter W. Jones and Thomas H. Wolff, Hausdorff dimension of harmonic measures in the plane, Acta Math. 161 (1988), no. 1-2, 131–144. MR 962097, DOI 10.1007/BF02392296
- Carlos E. Kenig and Tatiana Toro, Harmonic measure on locally flat domains, Duke Math. J. 87 (1997), no. 3, 509–551. MR 1446617, DOI 10.1215/S0012-7094-97-08717-2
- Carlos Kenig and Tatiana Toro, Free boundary regularity below the continuous threshold: 2-phase problems, J. Reine Angew. Math. 596 (2006), 1–44. MR 2254803, DOI 10.1515/CRELLE.2006.050
- Carlos E. Kenig and Tatiana Toro, Free boundary regularity for harmonic measures and Poisson kernels, Ann. of Math. (2) 150 (1999), no. 2, 369–454. MR 1726699, DOI 10.2307/121086
- John L. Lewis, Gregory C. Verchota, and Andrew L. Vogel, Wolff snowflakes, Pacific J. Math. 218 (2005), no. 1, 139–166. MR 2224593, DOI 10.2140/pjm.2005.218.139
- N. G. Makarov, On the distortion of boundary sets under conformal mappings, Proc. London Math. Soc. (3) 51 (1985), no. 2, 369–384. MR 794117, DOI 10.1112/plms/s3-51.2.369
- Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
- David Preiss, Geometry of measures in $\textbf {R}^n$: distribution, rectifiability, and densities, Ann. of Math. (2) 125 (1987), no. 3, 537–643. MR 890162, DOI 10.2307/1971410
- Leon Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR 756417
- Thomas H. Wolff, Counterexamples with harmonic gradients in $\textbf {R}^3$, Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991) Princeton Math. Ser., vol. 42, Princeton Univ. Press, Princeton, NJ, 1995, pp. 321–384. MR 1315554
Bibliographic Information
- C. Kenig
- Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
- MR Author ID: 100230
- Email: cek@math.uchicago.edu
- D. Preiss
- Affiliation: Mathematics Institut, University of Warwick, Coventry CV4 7AL, United Kingdom
- MR Author ID: 141890
- Email: d.preiss@warwick.ac.uk
- T. Toro
- Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195-4350.
- MR Author ID: 363909
- Email: toro@math.washington.edu
- Received by editor(s): October 25, 2007
- Published electronically: April 25, 2008
- Additional Notes: The first author was partially supported by NSF grant DMS-0456583.
The third author was partially supported by NSF grant DMS-0600915 - © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc. 22 (2009), 771-796
- MSC (2000): Primary 28A33, 31A15
- DOI: https://doi.org/10.1090/S0894-0347-08-00601-2
- MathSciNet review: 2505300