Regularity theory and traces of $\mathcal {A}$-harmonic functions
HTML articles powered by AMS MathViewer
- by Pekka Koskela, Juan J. Manfredi and Enrique Villamor PDF
- Trans. Amer. Math. Soc. 348 (1996), 755-766 Request permission
Abstract:
In this paper we discuss two different topics concerning $\mathcal {A}$- harmonic functions. These are weak solutions of the partial differential equation \begin{equation*}\text {div}(\mathcal {A}(x,\nabla u))=0,\end{equation*} where $\alpha (x)|\xi |^{p-1}\le \langle \mathcal {A}(x,\xi ),\xi \rangle \le \beta (x) |\xi |^{p-1}$ for some fixed $p\in (1,\infty )$, the function $\beta$ is bounded and $\alpha (x)>0$ for a.e. $x$. First, we present a new approach to the regularity of $\mathcal {A}$-harmonic functions for $p>n-1$. Secondly, we establish results on the existence of nontangential limits for $\mathcal {A}$-harmonic functions in the Sobolev space $W^{1,q}(\mathbb {B})$, for some $q>1$, where $\mathbb {B}$ is the unit ball in $\mathbb {R}^n$. Here $q$ is allowed to be different from $p$.References
- A. Beurling, Ensembles exceptionnels, Acta Math. 72 (1940), 1–13.
- B. Bojarski and T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in $\textbf {R}^{n}$, Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983), no. 2, 257–324. MR 731786, DOI 10.5186/aasfm.1983.0806
- 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
- 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, DOI 10.5186/aasfm.1980.0507
- S. Granlund, P. Lindqvist, and O. Martio, Conformally invariant variational integrals, Trans. Amer. Math. Soc. 277 (1983), no. 1, 43–73. MR 690040, DOI 10.1090/S0002-9947-1983-0690040-4
- 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
- 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
- L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63–97. MR 95, DOI 10.1007/BF02547750
- P. Lindqvist and J. Manfredi, The Harnack inequality for $\infty$-harmonic functions, Elec. J. Differential Equations, vol. 1995, n. 4, 1–5.
- Barkley Rosser, On the first case of Fermat’s last theorem, Bull. Amer. Math. Soc. 45 (1939), 636–640. MR 25, DOI 10.1090/S0002-9904-1939-07058-4
- J. J. Manfredi, Monotone Sobolev functions, J. Geom. Anal. 4 (1994), 393–402.
- J. J. Manfredi and E. Villamor, Traces of monotone Sobolev functions, J. Geom. Anal. (to appear).
- O. Martio and S. Rickman, Boundary behavior of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A. I. 507 (1972), 17. MR 379846
- 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
- Yoshihiro Mizuta, Boundary behavior of $p$-precise functions on a half space of $\textbf {R}^n$, Hiroshima Math. J. 18 (1988), no. 1, 73–94. MR 935884
- 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, DOI 10.5802/aif.952
- Alexander Nagel, Walter Rudin, and Joel H. Shapiro, Tangential boundary behavior of functions in Dirichlet-type spaces, Ann. of Math. (2) 116 (1982), no. 2, 331–360. MR 672838, DOI 10.2307/2007064
- Ju. G. Rešetnjak, The boundary behavior of functions with generalized derivatives, Sibirsk. Mat. Ž. 13 (1972), 411–419 (Russian). MR 0296687
- James Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247–302. MR 170096, DOI 10.1007/BF02391014
- Matti Vuorinen, Conformal geometry and quasiregular mappings, Lecture Notes in Mathematics, vol. 1319, Springer-Verlag, Berlin, 1988. MR 950174, DOI 10.1007/BFb0077904
- William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685, DOI 10.1007/978-1-4612-1015-3
Additional Information
- Pekka Koskela
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
- MR Author ID: 289254
- Email: pkoskela@math.jyu.fi
- Juan J. Manfredi
- Affiliation: Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
- MR Author ID: 205679
- Email: manfredit@pitt.edu
- Enrique Villamor
- Affiliation: Department of Mathematics, Florida International University, Miami, Florida 33199
- Email: villamor@fiu.edu
- 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 DMS-9305742
Research of the second author was partially supported by NSF grant DMS-9101864 - © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 348 (1996), 755-766
- MSC (1991): Primary 35B65; Secondary 31B25
- DOI: https://doi.org/10.1090/S0002-9947-96-01430-4
- MathSciNet review: 1311911