p Harmonic Measure in Simply Connected Domains
[Mesure p harmonique dans les régions simplement connexes]
Annales de l'Institut Fourier, Tome 61 (2011) no. 2, pp. 689-715.

Soit Ω une région bornée et simplement connexe dans le plan complexe . Soit N un voisinage de Ω. Pour 1<p<, on considère une solution positive p-harmonique faible u ^ de l’équation de p Laplace dans ΩN. Supposons que u ^ s’annule sur Ω au sens de Sobolev et qu’elle s’étend dans NΩ avec u ^0 en NΩ. Alors il existe une mesure positive finie de Borel μ ^ dans avec support contenu dans Ω telle que

|u^|p-2u^,φdA=-φdμ^

pour tout φC 0 (N). Si p=2 et si u ^ est la fonction de Green pour Ω avec pole xΩN ¯, alors la mesure μ ^ est la mesure harmonique au point x, ω=ω x , pour l’équation de Laplace. Dans ce travail on continue l’ étude commencée par le premier auteur, en établissant des nouveaux résultats, pour les régions simplement connexe, concernant la dimension de Hausdorff du support de la mesure μ ^. En particulier, on obtient des résultats, pour 1<p<, p2, qui rappèllent le fameux résultat de Makarov concernant la dimension de Hausdorff pour le support de la mesure harmonique des régions simplement connexes.

Let Ω be a bounded simply connected domain in the complex plane, . Let N be a neighborhood of Ω, let p be fixed, 1<p<, and let u ^ be a positive weak solution to the p Laplace equation in ΩN. Assume that u ^ has zero boundary values on Ω in the Sobolev sense and extend u ^ to NΩ by putting u ^0 on NΩ. Then there exists a positive finite Borel measure μ ^ on with support contained in Ω and such that

|u^|p-2u^,φdA=-φdμ^

whenever φC 0 (N). If p=2 and if u ^ is the Green function for Ω with pole at xΩN ¯ then the measure μ ^ coincides with harmonic measure at x, ω=ω x , associated to the Laplace equation. In this paper we continue the studies initiated by the first author by establishing new results, in simply connected domains, concerning the Hausdorff dimension of the support of the measure μ ^. In particular, we prove results, for 1<p<, p2, reminiscent of the famous result of Makarov concerning the Hausdorff dimension of the support of harmonic measure in simply connected domains.

Reçu le :
Accepté le :
DOI : https://doi.org/10.5802/aif.2626
Classification : 35J25,  35J70
Mots clés : fonction harmonique, mesure harmonique, mesure p harmonique, fonction p harmonique, région simplement connexe, mesure de Hausdorff, dimension de Hausdorff
@article{AIF_2011__61_2_689_0,
     author = {Lewis, John L. and Nystr\"om, Kaj and Poggi-Corradini, Pietro},
     title = {$p$ {Harmonic} {Measure} in {Simply} {Connected} {Domains}},
     journal = {Annales de l'Institut Fourier},
     pages = {689--715},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {61},
     number = {2},
     year = {2011},
     doi = {10.5802/aif.2626},
     zbl = {1241.35071},
     mrnumber = {2895070},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2626/}
}
Lewis, John L.; Nyström, Kaj; Poggi-Corradini, Pietro. $p$ Harmonic Measure in Simply Connected Domains. Annales de l'Institut Fourier, Tome 61 (2011) no. 2, pp. 689-715. doi : 10.5802/aif.2626. https://aif.centre-mersenne.org/articles/10.5802/aif.2626/

[1] Aikawa, H.; Shanmugalingam, N. Carleson-type estimates for p - harmonic functions and the conformal Martin boundary of John domains in metric measure spaces, Michigan Math. J., Volume 53 (2005), pp. 165-188 | Article | MR 2125540 | Zbl 1076.31006

[2] Balogh, Z.; Bonk, M. Lengths of radii under conformal maps of the unit disc, Proc.  Amer.  Math.  Soc., Volume 127 (1999) no. 3, pp. 801-804 | Article | MR 1469396 | Zbl 0939.30018

[3] Batakis, A. Harmonic measure of some Cantor type sets, Ann. Acad. Sci. Fenn. Math., Volume 21 (1996) no. 2, pp. 255-270 | MR 1404086 | Zbl 0849.31005

[4] Bennewitz, B.; Lewis, J. On the dimension of p-harmonic measure, Ann. Acad. Sci. Fenn. Math., Volume 30 (2005) no. 2, pp. 459-505 | MR 2173375 | Zbl 1194.35189

[5] Bourgain, J. On the Hausdorff dimension of harmonic measure in higher dimensions, Inv. Math., Volume 87 (1987), pp. 477-483 | Article | MR 874032 | Zbl 0616.31004

[6] Caffarelli, L.; Fabes, E.; Mortola, S.; Salsa, S. Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana J. Math., Volume 30 (1981) no. 4, pp. 621-640 | Article | MR 620271 | Zbl 0512.35038

[7] Carleson, L. On the existence of boundary values for harmonic functions in several variables, Ark. Mat., Volume 4 (1962), pp. 393-399 | Article | MR 159013 | Zbl 0107.08402

[8] Carleson, L. On the support of harmonic measure for sets of Cantor type, Ann. Acad. Sci. Fenn., Volume 10 (1985), pp. 113-123 | MR 802473 | Zbl 0593.31004

[9] Domar, Y. On the existence of a largest subharmonic minorant of a given function, Ark. Mat., Volume 3 (1957), pp. 429-440 | Article | MR 87767 | Zbl 0078.09301

[10] Garnett, J.; Marshall, D. Harmonic Measure, Cambridge University Press, 2005 | MR 2150803 | Zbl 1139.31001

[11] Hedenmalm, H.; Kayamov, I. On the Makarov law of the iterated logarithm, Proc. Amer. Math. Soc., Volume 135 (2007) no. 7, pp. 2235-2248 | Article | MR 2299501 | Zbl 1117.30024

[12] Heinonen, J.; Kilpeläinen, T.; Martio, O. Nonlinear potential theory of degenerate elliptic equations, Oxford University Press, 1993 | MR 1207810 | Zbl 0780.31001

[13] Jones, P.; Wolff, T. Hausdorff dimension of harmonic measure in the plane, Acta Math., Volume 161 (1988), pp. 131-144 | Article | MR 962097 | Zbl 0667.30020

[14] Kaufmann, R.; Wu, J.M. On the snowflake domain, Ark. Mat., Volume 23 (1985), pp. 177-183 | Article | MR 800179 | Zbl 0573.30030

[15] Lewis, J. Note on p harmonic measure, Computational Methods in Function Theory, Volume 6 (2006) no. 1, pp. 109-144 | MR 2241036 | Zbl 1161.35406

[16] Lewis, J.; Nyström, K. Boundary Behaviour and the Martin Boundary Problem for p-Harmonic Functions in Lipschitz domains (to appear in Annals of Mathematics) | Zbl 1210.31004

[17] Lewis, J.; Nyström, K. Boundary Behaviour of p-Harmonic Functions in Domains Beyond Lipschitz Domains, Advances in the Calculus of Variations, Volume 1 (2008), pp. 133-177 | Article | MR 2427450 | Zbl 1169.31004

[18] Lewis, J.; Nyström, K. Regularity and Free Boundary Regularity for the p-Laplacian in Lipschitz and C 1 -domains, Annales Acad. Sci. Fenn. Mathematica, Volume 33 (2008), pp. 523-548 | MR 2431379 | Zbl 1202.35110

[19] Lewis, J.; Nyström, K. Boundary Behaviour for p-Harmonic Functions in Lipschitz and Starlike Lipschitz Ring Domains, Annales Scientifiques de L’Ecole Normale Superieure, Volume 40 (September-October 2007) no. 5, pp. 765-813 | Article | Numdam | MR 2382861 | Zbl 1134.31008

[20] Lewis, J.; Verchota, G.; Vogel, A. On Wolff snowflakes, Pacific Journal of Mathematics, Volume 218 (2005), pp. 139-166 | Article | MR 2224593 | Zbl 1108.31006

[21] Makarov, N. Distortion of boundary sets under conformal mapping, Proc. London Math. Soc., Volume 51 (1985), pp. 369-384 | Article | MR 794117 | Zbl 0573.30029

[22] Mattila, P. Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, 1995 | MR 1333890 | Zbl 0819.28004

[23] Pommerenke, C. Univalent Functions, Vandenhoeck and Ruprecht, Göttingen, 1975 | MR 507768 | Zbl 0298.30014

[24] Volberg, A. On the dimension of harmonic measure of Cantor repellers, Michigan Math. J., Volume 40 (1993), pp. 239-258 | Article | MR 1226830 | Zbl 0797.30022

[25] Wolff, T. Plane harmonic measures live on sets of finite linear length, Ark. Mat., Volume 31 (1993) no. 1, pp. 137-172 | Article | MR 1230270 | Zbl 0809.30007

[26] Wolff, Thomas H. Counterexamples with harmonic gradients in R 3 , Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991) (Princeton Math. Ser.) Volume 42, Princeton Univ. Press, Princeton, NJ, 1995, pp. 321-384 | MR 1315554 | Zbl 0836.31004