Geodesic
On Liouville theorems for harmonic functions with finite Dirichlet integral
Sbornik. Mathematics, Tome 60 (1988) no. 2, pp. 485-504 Cet article a éte moissonné depuis la source Math-Net.Ru

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A criterion for the validity of the $D$-Liouville theorem is proved. In § 1 it is shown that the question of $L^\infty$ and $D$-Liouville theorems reduces to the study of the so-called massive sets (in other words, the level sets of harmonic functions in the classes $L^\infty$ and $L^\infty\cap D$). In § 2 some properties of capacity are presented. In § 3 the criterion of $D$-massiveness is formulated – the central result of this article – and examples are presented. In § 4 a criterion for the $D$-Liouville theorem is formulated, and corollaries are derived. §§ 5–9 the main theorems are proved. Figures: 5. Bibliography: 17 titles. 
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     title = {On {Liouville} theorems for harmonic functions with finite {Dirichlet} integral},
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     year = {1988},
     volume = {60},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1988_60_2_a14/}
}
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A. A. Grigor'yan. On Liouville theorems for harmonic functions with finite Dirichlet integral. Sbornik. Mathematics, Tome 60 (1988) no. 2, pp. 485-504. http://geodesic.mathdoc.fr/item/SM_1988_60_2_a14/

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