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Title: Pointwise estimates of nonnegative subsolutions of quasilinear elliptic equations at irregular boundary points (English)
Author: Malý, Jan
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 37
Issue: 1
Year: 1996
Pages: 23-42
Category: math
Summary: Let $u$ be a weak solution of a quasilinear elliptic equation of the growth $p$ with a measure right hand term $\mu$. We estimate $u(z)$ at an interior point $z$ of the domain $\Omega$, or an irregular boundary point $z\in \partial\Omega$, in terms of a norm of $u$, a nonlinear potential of $\mu$ and the Wiener integral of $\bold R^n\setminus \Omega$. This quantifies the result on necessity of the Wiener criterion. (English)
Keyword: elliptic equations
Keyword: Wiener criterion
Keyword: nonlinear potentials
Keyword: measure data
MSC: 35B45
MSC: 35D05
MSC: 35J65
MSC: 35J67
MSC: 35J70
MSC: 35R05
idZBL: Zbl 0851.35047
idMR: MR1396160
Date available: 2009-01-08T18:22:05Z
Last updated: 2012-04-30
Stable URL:
Reference: [1] Adams D.R.: $L^p$ potential theory techniques and nonlinear PDE.In: Potential Theory (Ed. M. Kishi) Walter de Gruyter & Co Berlin (1992), 1-15. Zbl 0760.22013, MR 1167217
Reference: [2] Adams D.R., Hedberg L.I.: Function Spaces and Potential Theory.Springer Verlag Berlin (1995). Zbl 0834.46021, MR 1411441
Reference: [3] Adams D.R., Meyers N.G.: Thinness and Wiener criteria for non-linear potentials.Indiana Univ. Math. J. 22 (1972), 169-197. Zbl 0244.31012, MR 0316724
Reference: [4] Brelot M.: On Topologies and Boundaries in Potential Theory.Lecture Notes in Math. 175, Springer ({1971}). Zbl 0222.31014, MR 0281940
Reference: [5] Federer H., Ziemer W.P.: The Lebesgue set of a function whose partial derivatives are $p$-th power summable.Indiana Univ. Math. J. 22 (1972), 139-158. MR 0435361
Reference: [6] Frehse J.: Capacity methods in the theory of partial differential equations.Jahresber. Deutsch. Math. Verein. 84 (1982), 1-44. Zbl 0486.35002, MR 0644068
Reference: [7] Fuglede B.: The quasi topology associated with a countably subadditive set function.Ann. Inst. Fourier Grenoble 21.1 (1971), 123-169. Zbl 0197.19401, MR 0283158
Reference: [8] Gariepy R., Ziemer W.P.: A regularity condition at the boundary for solutions of quasilinear elliptic equations.Arch. Rat. Mech. Anal. 67 (1977), 25-39. Zbl 0389.35023, MR 0492836
Reference: [9] Hedberg L.I.: Nonlinear potentials and approximation in the mean by analytic functions.Math. Z. 129 (1972), 299-319. MR 0328088
Reference: [10] Hedberg L.I., Wolff Th.H.: Thin sets in nonlinear potential theory.Ann. Inst. Fourier 33.4 (1983), 161-187. Zbl 0508.31008, MR 0727526
Reference: [11] Heinonen J., Kilpeläinen T.: On the Wiener criterion and quasilinear obstacle problems.Trans. Amer. Math. Soc. 310 (1988), 239-255. MR 0965751
Reference: [12] Heinonen J., Kilpeläinen T., Martio O.: Fine topology and quasilinear elliptic equations.Ann. Inst. Fourier 39.2 (1989), 293-318. MR 1017281
Reference: [13] Heinonen J., Kilpeläinen T., Martio O.: Nonlinear Potential Theory of Degenerate Elliptic Equations.Oxford University Press, Oxford (1993). MR 1207810
Reference: [14] Kilpeläinen T., Malý J.: Degenerate elliptic equations with measure data and nonlinear potentials.Ann. Scuola Norm. Sup. Pisa. Cl. Science, Ser. IV 19 (1992), 591-613. MR 1205885
Reference: [15] Kilpeläinen T., Malý J.: Supersolutions to degenerate elliptic equations on quasi open sets.Comm. Partial Differential Equations 17 (1992), 371-405. MR 1163430
Reference: [16] Kilpeläinen T., Malý J.: The Wiener test and potential estimates for quasilinear elliptic equations.Acta Math. 172 (1994), 137-161. MR 1264000
Reference: [17] Lieberman G.M.: Sharp forms of estimates for subsolutions and supersolutions of quasilinear elliptic equations with right hand side a measure.Comm. Partial Differential Equations 18 (1993), 1991-2112. MR 1233190
Reference: [18] Lindqvist P., Martio O.: Two theorems of N. Wiener for solutions of quasilinear elliptic equations.Acta Math. 155 (1985), 153-171. Zbl 0607.35042, MR 0806413
Reference: [19] Littman W., Stampacchia G., Weinberger H.F.: Regular points for elliptic equations with discontinuous coefficients.Ann. Scuola Norm. Sup. Pisa. Serie III 17 (1963), 43-77. Zbl 0116.30302, MR 0161019
Reference: [20] Malý J.: Nonlinear potentials and quasilinear PDE's.Proceedings of the International Conference on Potential Theory, Kouty, 1994, to appear. Zbl 0857.35046, MR 1404703
Reference: [21] Maz'ya V.G.: On the continuity at a boundary point of solutions of quasi-linear elliptic equations (Russian).Vestnik Leningrad. Univ. 25 42-55 English translation Vestnik Leningrad. Univ. Math. 3 (1976), 225-242. MR 0274948
Reference: [22] Maz'ya V.G., Khavin V.P.: Nonlinear potential theory (Russian).Uspekhi Mat. Nauk 27.6 (1972), 67-138 English translation Russian Math. Surveys 27 (1972), 71-148.
Reference: [23] Malý J., Ziemer W.P.: Fine Regularity of Solutions of Elliptic in preparation.
Reference: [24] Meyers N.G.: Continuity properties of potentials.Duke Math. J. 42 (1975), 157-166. Zbl 0334.31004, MR 0367235
Reference: [25] Rakotoson J.M., Ziemer W.P.: Local behavior of solutions of quasilinear elliptic equations with general structure.Trans. Amer. Math. Soc. 319 (1990), 747-764. Zbl 0708.35023, MR 0998128
Reference: [26] Skrypnik I.V.: Nonlinear Elliptic Boundary Value Problems.Teubner Verlag, Leipzig (1986). Zbl 0617.35001, MR 0915342
Reference: [27] Trudinger N.S.: On Harnack type inequalities and their application to quasilinear elliptic equations.Comm. Pure Appl. Math. 20 (1967), 721-747. Zbl 0153.42703, MR 0226198
Reference: [28] Wiener N.: Certain notions in potential theory.J. Math. Phys. 3 (1924), 24-5 Reprinted in: Norbert Wiener: Collected works. Vol. 1 (1976), MIT Press, pp. 364-391. MR 0532698


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