Nakai, Mitsuru Existence of Dirichlet finite harmonic measures on Euclidean balls. (English) Zbl 0794.31007 Nagoya Math. J. 133, 85-125 (1994). The author calls a function \(w\) an \({\mathcal A}\)-harmonic measure in the unit ball \(B\) of \(\mathbb{R}^ n\) if \(w\) is \({\mathcal A}\)-harmonic in \(B\) and if the greatest \({\mathcal A}\)-harmonic minorant of \(\min(w,1-w)\) is the zero function; by an \({\mathcal A}\)-harmonic function \(u\) we mean a solution of the equation \(-\text{div } {\mathcal A}(x, \nabla u)=0\) in \(B\). The typical model equation is the \(p\)-Laplacian \(-\text{div } (|\nabla u|^{p-2} \nabla u)=0\), where \(1<p\leq n\); see the recent monograph by J. Heinonen, the reviewer and O. Martio [Nonlinear potential theory of degenerate elliptic equations (1993; Zbl 0780.31001)] for the potential theory of \({\mathcal A}\)-harmonic functions.In the paper under review the author establishes that there exists a nonconstant \({\mathcal A}\)-harmonic measure \(w\) in \(B\) with finite Dirichlet integral, \[ \int_{B} |\nabla w|^ p dx<\infty, \] exactly when \(1<p<2\). Further, this is shown to be equivalent to the disconnectivity of the Royden harmonic boundary with exponent \(p\) of \(B\). Similar results are proved by D. A. Herron and P. Koskela in their recent manuscript “Continuity of Sobolev functions and Dirichlet finite harmonic measures”.The paper is rather self-contained: no famililarity with Sobolev spaces techniques nor with nonlinear potential theory is assumed. Reviewer: T.Kilpeläinen (Jyväskylä) Cited in 1 ReviewCited in 2 Documents MSC: 31C45 Other generalizations (nonlinear potential theory, etc.) 35J70 Degenerate elliptic equations Keywords:harmonic measures; nonlinear potential theory; \(p\)-Royden boundary Citations:Zbl 0780.31001 PDFBibTeX XMLCite \textit{M. Nakai}, Nagoya Math. J. 133, 85--125 (1994; Zbl 0794.31007) Full Text: DOI References: [1] DOI: 10.1016/0362-546X(90)90135-4 · Zbl 0712.31004 [2] DOI: 10.2140/pjm.1972.40.397 · Zbl 0241.31014 [3] DOI: 10.1007/BF02391778 · Zbl 0173.39202 [4] Proceedings of the International Conference on Functional Analysis and Related Topics pp 316– (1969) [5] Sobolev Spaces (1985) · Zbl 0727.46017 [6] Vestnik Leningrad Univ. 3 pp 225– (1976) [7] The Theory of Partial Differential Equations (1973) [8] Topology from the Differentiable Viewpoint (1965) · Zbl 0136.20402 [9] Ann. Acad. Sci. Fenn., Ser. AI Math. Dissertation 74 pp 1– (1990) [10] DOI: 10.1090/S0002-9947-1971-0281912-5 [11] DOI: 10.2307/1969809 · Zbl 0065.31102 [12] DOI: 10.1215/S0012-7094-73-04016-7 · Zbl 0272.30025 [13] Nagoya Math. J. 46 pp 1– (1972) [14] J. Math. Kyoto Univ. 5 pp 197– (1966) · Zbl 0156.09001 [15] Linear Operators, Part I (1967) [16] Function Theory of Several Complex Variables (1982) [17] Variété, Différentiables (1955) [18] J. Math. Kyoto Univ. 29 pp 625– (1989) · Zbl 0721.30030 [19] Ideale Ränder Riemannscher Flächen (1963) [20] (1960) [21] Sobolev Spaces (1978) [22] DOI: 10.1090/S0002-9947-1960-0125216-2 [23] DOI: 10.1007/BF02391014 · Zbl 0128.09101 [24] Classification Theory of Riemann Surfaces (1970) · Zbl 0199.40603 [25] Real and Complex Analysis (1987) · Zbl 0925.00005 [26] Amer. Math. Soc (1989) [28] DOI: 10.2996/kmj/1138036489 · Zbl 0501.30017 [29] DOI: 10.1080/17476939308814619 · Zbl 0795.31001 [30] DOI: 10.1215/S0012-7094-92-06725-1 · Zbl 0764.53032 [31] DOI: 10.1080/17476939108814503 · Zbl 0724.30034 [32] DOI: 10.1512/iumj.1991.40.40050 · Zbl 0753.30033 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.