Geodesic
Boundary structure and size in terms of interior and exterior harmonic measures in higher dimensions
Kenig, C.1 ; Preiss, D.2 ; Toro, T.3
1 Department of Mathematics, University of Chicago, Chicago, Illinois 60637
2 Mathematics Institut, University of Warwick, Coventry CV4 7AL, United Kingdom
3 Department of Mathematics, University of Washington, Seattle, Washington 98195-4350.
Journal of the American Mathematical Society, Tome 22 (2009) no. 3, pp. 771-796

Voir la notice de l'article provenant de la source American Mathematical Society

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 $n\ge 3$.
DOI : 10.1090/S0894-0347-08-00601-2

Kenig, C. 1 ; Preiss, D. 2 ; Toro, T. 3

1 Department of Mathematics, University of Chicago, Chicago, Illinois 60637
2 Mathematics Institut, University of Warwick, Coventry CV4 7AL, United Kingdom
3 Department of Mathematics, University of Washington, Seattle, Washington 98195-4350. 
@article{10_1090_S0894_0347_08_00601_2,
     author = {Kenig, C. and Preiss, D. and Toro, T.},
     title = {Boundary structure and size in terms of interior and exterior harmonic measures in higher dimensions},
     journal = {Journal of the American Mathematical Society},
     pages = {771--796},
     publisher = {mathdoc},
     volume = {22},
     number = {3},
     year = {2009},
     doi = {10.1090/S0894-0347-08-00601-2},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-08-00601-2/}
}
TY  - JOUR
AU  - Kenig, C.
AU  - Preiss, D.
AU  - Toro, T.
TI  - Boundary structure and size in terms of interior and exterior harmonic measures in higher dimensions
JO  - Journal of the American Mathematical Society
PY  - 2009
SP  - 771
EP  - 796
VL  - 22
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-08-00601-2/
DO  - 10.1090/S0894-0347-08-00601-2
ID  - 10_1090_S0894_0347_08_00601_2
ER  - 
%0 Journal Article
%A Kenig, C.
%A Preiss, D.
%A Toro, T.
%T Boundary structure and size in terms of interior and exterior harmonic measures in higher dimensions
%J Journal of the American Mathematical Society
%D 2009
%P 771-796
%V 22
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-08-00601-2/
%R 10.1090/S0894-0347-08-00601-2
%F 10_1090_S0894_0347_08_00601_2
Kenig, C.; Preiss, D.; Toro, T. Boundary structure and size in terms of interior and exterior harmonic measures in higher dimensions. Journal of the American Mathematical Society, Tome 22 (2009) no. 3, pp. 771-796. doi: 10.1090/S0894-0347-08-00601-2

[1] Alt, Hans Wilhelm, Caffarelli, Luis A., Friedman, Avner Variational problems with two phases and their free boundaries Trans. Amer. Math. Soc. 1984 431 461

[2] Bishop, Christopher J. Some questions concerning harmonic measure 1992 89 97

[3] Bishop, C. J., Carleson, L., Garnett, J. B., Jones, P. W. Harmonic measures supported on curves Pacific J. Math. 1989 233 236

[4] Brothers, John E., Ziemer, William P. Minimal rearrangements of Sobolev functions J. Reine Angew. Math. 1988 153 179

[5] Carleson, Lennart On the support of harmonic measure for sets of Cantor type Ann. Acad. Sci. Fenn. Ser. A I Math. 1985 113 123

[6] Evans, Lawrence C., Gariepy, Ronald F. Measure theory and fine properties of functions 1992

[7] Falconer, Kenneth Techniques in fractal geometry 1997

[8] Garnett, John B., Marshall, Donald E. Harmonic measure 2005

[9] Hardt, Robert, Simon, Leon Nodal sets for solutions of elliptic equations J. Differential Geom. 1989 505 522

[10] Jones, Peter W. A geometric localization theorem Adv. in Math. 1982 71 79

[11] Jones, Peter W., Wolff, Thomas H. Hausdorff dimension of harmonic measures in the plane Acta Math. 1988 131 144

[12] Kenig, Carlos E., Toro, Tatiana Harmonic measure on locally flat domains Duke Math. J. 1997 509 551

[13] Kenig, Carlos, Toro, Tatiana Free boundary regularity below the continuous threshold: 2-phase problems J. Reine Angew. Math. 2006 1 44

[14] Kenig, Carlos E., Toro, Tatiana Free boundary regularity for harmonic measures and Poisson kernels Ann. of Math. (2) 1999 369 454

[15] Lewis, John L., Verchota, Gregory C., Vogel, Andrew L. Wolff snowflakes Pacific J. Math. 2005 139 166

[16] Makarov, N. G. On the distortion of boundary sets under conformal mappings Proc. London Math. Soc. (3) 1985 369 384

[17] Mattila, Pertti Geometry of sets and measures in Euclidean spaces 1995

[18] Preiss, David Geometry of measures in 𝑅ⁿ: distribution, rectifiability, and densities Ann. of Math. (2) 1987 537 643

[19] Simon, Leon Lectures on geometric measure theory 1983

[20] Wolff, Thomas H. Counterexamples with harmonic gradients in 𝑅³ 1995 321 384

Cité par Sources :