Regularity theory and traces of harmonic functions
Authors:
Pekka Koskela, Juan J. Manfredi and Enrique Villamor
Journal:
Trans. Amer. Math. Soc. 348 (1996), 755766
MSC (1991):
Primary 35B65; Secondary 31B25
MathSciNet review:
1311911
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: In this paper we discuss two different topics concerning  harmonic functions. These are weak solutions of the partial differential equation where for some fixed , the function is bounded and for a.e. . First, we present a new approach to the regularity of harmonic functions for . Secondly, we establish results on the existence of nontangential limits for harmonic functions in the Sobolev space , for some , where is the unit ball in . Here is allowed to be different from .
 [B]
A. Beurling, Ensembles exceptionnels, Acta Math. 72 (1940), 113.
 [BI]
B.
Bojarski and T.
Iwaniec, Analytical foundations of the theory of quasiconformal
mappings in 𝑅ⁿ, Ann. Acad. Sci. Fenn. Ser. A I Math.
8 (1983), no. 2, 257–324. MR 731786
(85h:30023)
 [C]
Lennart
Carleson, Selected problems on exceptional sets, Van Nostrand
Mathematical Studies, No. 13, D. Van Nostrand Co., Inc., Princeton,
N.J.Toronto, Ont.London, 1967. MR 0225986
(37 #1576)
 [G]
Seppo
Granlund, Harnack’s inequality in the borderline case,
Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1,
159–163. MR
595186 (81m:35041), http://dx.doi.org/10.5186/aasfm.1980.0507
 [GLM]
S.
Granlund, P.
Lindqvist, and O.
Martio, Conformally invariant variational
integrals, Trans. Amer. Math. Soc.
277 (1983), no. 1,
43–73. MR
690040 (84f:30030), http://dx.doi.org/10.1090/S00029947198306900404
 [HKM]
Juha
Heinonen, Tero
Kilpeläinen, and Olli
Martio, Nonlinear potential theory of degenerate elliptic
equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford
University Press, New York, 1993. Oxford Science Publications. MR 1207810
(94e:31003)
 [H]
Ilkka
Holopainen, Nonlinear potential theory and quasiregular mappings on
Riemannian manifolds, Ann. Acad. Sci. Fenn. Ser. A I Math.
Dissertationes 74 (1990), 45. MR 1052971
(91e:31029)
 [KK]
T. Kilpeläinen and P. Koskela, Global integrability of gradients of solutions to partial differential equations, Nonlinear Anal. 23 (1994), 899909, MR 95:03.
 [LM]
P. Lindqvist and J. Manfredi, The Harnack inequality for harmonic functions, Elec. J. Differential Equations, vol. 1995, n. 4, 15.
 [ML]
G. R. MacLane, Holomorphic functions, of arbitrarily slow growth, without radial limits, Michigan Math. J. 9 (1962), 2124, MR 25:203.
 [M]
J. J. Manfredi, Monotone Sobolev functions, J. Geom. Anal. 4 (1994), 393402.
 [MV]
J. J. Manfredi and E. Villamor, Traces of monotone Sobolev functions, J. Geom. Anal. (to appear).
 [MR]
O.
Martio and S.
Rickman, Boundary behavior of quasiregular mappings, Ann.
Acad. Sci. Fenn. Ser. A I 507 (1972), 17. MR 0379846
(52 #751)
 [Mi1]
Yoshihiro
Mizuta, Existence of various boundary limits of Beppo Levi
functions of higher order, Hiroshima Math. J. 9
(1979), no. 3, 717–745. MR 549670
(81d:31013)
 [Mi2]
Yoshihiro
Mizuta, Boundary behavior of 𝑝precise functions on a half
space of 𝑅ⁿ, Hiroshima Math. J. 18
(1988), no. 1, 73–94. MR 935884
(89d:31014)
 [Mi3]
Yoshihiro
Mizuta, On the boundary limits of harmonic functions with gradient
in 𝐿^{𝑝}, Ann. Inst. Fourier (Grenoble)
34 (1984), no. 1, 99–109 (English, with French
summary). MR
743623 (85f:31009)
 [NRS]
Alexander
Nagel, Walter
Rudin, and Joel
H. Shapiro, Tangential boundary behavior of functions in
Dirichlettype spaces, Ann. of Math. (2) 116 (1982),
no. 2, 331–360. MR 672838
(84a:31002), http://dx.doi.org/10.2307/2007064
 [R]
Ju.
G. Rešetnjak, The boundary behavior of functions with
generalized derivatives, Sibirsk. Mat. Ž. 13
(1972), 411–419 (Russian). MR 0296687
(45 #5746)
 [S]
James
Serrin, Local behavior of solutions of quasilinear equations,
Acta Math. 111 (1964), 247–302. MR 0170096
(30 #337)
 [V]
Matti
Vuorinen, Conformal geometry and quasiregular mappings,
Lecture Notes in Mathematics, vol. 1319, SpringerVerlag, Berlin,
1988. MR
950174 (89k:30021)
 [Z]
William
P. Ziemer, Weakly differentiable functions, Graduate Texts in
Mathematics, vol. 120, SpringerVerlag, New York, 1989. Sobolev spaces
and functions of bounded variation. MR 1014685
(91e:46046)
 [B]
 A. Beurling, Ensembles exceptionnels, Acta Math. 72 (1940), 113.
 [BI]
 B. Bojarski and T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in , Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983), 257324, MR 85h:30023.
 [C]
 L. Carleson, Selected problems in exceptional sets, Van Nostrand Math. Studies, vol. 13, Van Nostrand, Princeton, NJ, 1967, MR 37:1576.
 [G]
 S. Granlund, Harnack's inequality in the borderline case, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), 159164, MR 81m:35041.
 [GLM]
 S. Granlund, P. Lindqvist, and O. Martio, Conformally invariant variational integrals, Trans. Amer. Math. Soc. 277 (1983), 4373, MR 84f:30030.
 [HKM]
 J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear potential theory, Oxford Univ. Press, Oxford, 1993, MR 94e:31003.
 [H]
 I. Holopainen, Nonlinear potential theory and quasiregular mappings on Riemannian manifolds, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 74 (1990), 145, MR 91e:31029.
 [KK]
 T. Kilpeläinen and P. Koskela, Global integrability of gradients of solutions to partial differential equations, Nonlinear Anal. 23 (1994), 899909, MR 95:03.
 [LM]
 P. Lindqvist and J. Manfredi, The Harnack inequality for harmonic functions, Elec. J. Differential Equations, vol. 1995, n. 4, 15.
 [ML]
 G. R. MacLane, Holomorphic functions, of arbitrarily slow growth, without radial limits, Michigan Math. J. 9 (1962), 2124, MR 25:203.
 [M]
 J. J. Manfredi, Monotone Sobolev functions, J. Geom. Anal. 4 (1994), 393402.
 [MV]
 J. J. Manfredi and E. Villamor, Traces of monotone Sobolev functions, J. Geom. Anal. (to appear).
 [MR]
 O. Martio and S. Rickman, Boundary behavior of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 507 (1972), 117, MR 52:751.
 [Mi1]
 Y. Mizuta, Existence of various boundary limits of Beppo Levi functions of higher order, Hiroshima Math. J. 9 (1979), 717745, MR 81d:31013.
 [Mi2]
 , Boundary behavior of precise functions on a half space of , Hiroshima Math. J. 18 (1988), 7394, MR 89d:31014.
 [Mi3]
 , On the boundary limits of harmonic functions with gradient in , Ann. Inst. Fourier (Grenoble) 34 (1984), 99109, MR 85f:31009.
 [NRS]
 A. Nagel, W. Rudin, and J. H. Shapiro, Tangential boundary behavior of functions in Dirichlettype spaces, Ann. of Math. (2) 116 (1982), 331360, MR 84a:31002.
 [R]
 Yu. G. Resetnyak, Boundary behavior of functions with generalized derivatives, Siberian Math. J. 13 (1972), 285290, MR 45:5746.
 [S]
 J. Serrin, Local behavior of solutions of quasilinear equations, Acta Math. 111 (1964), 247302, MR 30:337.
 [V]
 M. Vuorinen, Conformal geometry and quasiregular mappings, Lecture Notes in Math., vol. 1319, SpringerVerlag, Berlin and New York, 1988, MR 89k:30021.
 [Z]
 W. Ziemer, Weakly differentiable functions, Graduate Texts in Math., vol. 120, Springer
Verlag, Berlin and New York, 1989, MR 91e:46046.
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Additional Information
Pekka Koskela
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email:
pkoskela@math.jyu.fi
Juan J. Manfredi
Affiliation:
Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email:
manfredit@pitt.edu
Enrique Villamor
Affiliation:
Department of Mathematics, Florida International University, Miami, Florida 33199
Email:
villamor@fiu.edu
DOI:
http://dx.doi.org/10.1090/S0002994796014304
PII:
S 00029947(96)014304
Received by editor(s):
June 7, 1994
Received by editor(s) in revised form:
January 23, 1995
Additional Notes:
Research of the first author was partially supported by the Academy of Finland and NSF grant DMS9305742
Research of the second author was partially supported by NSF grant DMS9101864
Article copyright:
© Copyright 1996
American Mathematical Society
