×

Boundary behavior of monotone Sobolev functions on John domains in a metric space. (English) Zbl 1074.31005

The authors study weighted boundary limits of monotone Sobolev functions on bounded \((\eta,\varphi)\)-John domains in a metric space. The result obtained in the paper extends some results by the second author [Hiroshima Math. J. 18, No. 1, 207–217 (1988; Zbl 0664.31007), Ann. Inst. Fourier 40, No. 4, 811–833 (1990; Zbl 0715.31002), Complex Variables, Theory Appl. 27, No. 2, 117–131 (1995; Zbl 0845.31002), Ann. Acad. Sci. Fenn., Ser. A I, Math. 20, No. 2, 315–326 (1995; Zbl 0852.31008)] for harmonic functions, polyharmonic functions and monotone functions on the upper half space \(\mathbb{R}^n_+\).

MSC:

31B25 Boundary behavior of harmonic functions in higher dimensions
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lebesgue, H.. Sur le probléme de Dirichlet Rendiconti del Circolo Matematico di Palermo, 1907; 24: 371–402
[2] Mizuta, Y.. Potential Theory in Euclidean Spaces, 1996 · Zbl 0849.31001
[3] Vuorinen, M.. Conformal Geometry and Quasiregular Mappings, 1988; Vol. 1319: Lectures Notes in Math. · Zbl 0646.30025
[4] Heinonen, J., Kilpeläinen, T., Martio, O.. Nonlinear Potential Theory of Degenerate Elliptic Equations, 1993 · Zbl 0780.31001
[5] Hajłasz, P., Koskela, P.. Sobolev met Poincaré Memoirs of the American Mathematical Society, 2000; 145: · Zbl 0954.46022
[6] Maz’ya, V.G., Poborichi, S.V.. Differentiable Functions on Bad Domains, 1997
[7] Mizuta, Y.. On the limits of harmonic functions Hiroshima Mathematical Journal, 1987; 18: 207–217 · Zbl 0664.31007
[8] Mizuta, Y.. On the existence of weighted boundary limits of harmonic functions Annales de l’Institut Fourier, 1990; 40: 811–833 · Zbl 0715.31002
[9] Mizuta, Y.. Boundary limits of polyharmonic functions in Sobolev-Orlicz spaces Complex Variables, 1995; 27: 117–131 · Zbl 0845.31002
[10] Mizuta, Y.. Tangential limits of monotone Sobolev functions Annales Academiae Scientiarium Fennicae. Ser. A. I. Math., 1995; 20: 315–326 · Zbl 0852.31008
[11] Väisälä, J.. Uniform domains The Tohoku Mathematical Journal, 1988; 40: 101–118 · Zbl 0627.30017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.