scholarly journals Rough isometries and $p$-harmonic functions with finite Dirichlet integral

1994 ◽  
pp. 143-176 ◽  
Author(s):  
Ilkka Holopainen
Keyword(s):  
Harmonic Functions ◽  
Dirichlet Integral

scholarly journals On the existence of various bounded harmonic functions with given periods, II

1975 ◽  
Vol 56 ◽  
pp. 1-5
Author(s):  
Masaru Hara
Keyword(s):  
Riemann Surface ◽  
Harmonic Function ◽  
Harmonic Functions ◽  
Dirichlet Integral ◽  
Period Function ◽  
Boundedness Property ◽  
One Dimensional ◽  
Bounded Harmonic Functions

Given a harmonic function u on a Riemann surface R, we define a period functionfor every one-dimensional cycle γ of the Riemann surface R. Γx(R) denote the totality of period functions Γu such that harmonic functions u satisfy a boundedness property X. As for X, we let B stand for boundedness, and D for the finiteness of the Dirichlet integral.

A Maximum Principle for Bounded Harmonic Functions on Riemannian Spaces

1970 ◽  
Vol 22 (4) ◽  
pp. 847-854 ◽  
Author(s):  
Y. K. Kwon ◽  
L. Sario
Keyword(s):  
Harmonic Functions ◽  
Riemann Surfaces ◽  
Higher Dimensions ◽  
Classification Theory ◽  
Superharmonic Function ◽  
Maximum Principles ◽  
Topological Properties ◽  
Positive Continuous Function ◽  
Dirichlet Integral ◽  
Bounded Harmonic Functions

Harmonic functions with certain boundedness properties on a given open Riemann surface R attain their maxima and minima on the harmonic boundary ΔB of R. The significance of such maximum principles lies in the fact that the classification theory of Riemann surfaces related to harmonic functions reduces to a study of topological properties of Δ(cf. [11; 8; 3; 12].For the corresponding problem in higher dimensions we shall first show that the complement of ΔR with respect to the Royden boundary ΓR of a Riemannian N-space R is harmonically negligible: given any non-empty compact subset E of ΓR – ΔR there exists an Evans superharmonic function v, i.e., a positive continuous function on R* = R ∪ ΓR, superharmonic on R, with v = 0 on ΔR, v ≡ ∞ on E, and with a finite Dirichlet integral over R.

Rough isometry and energy-finite solutions for the Schrödinger operator on Riemannian manifolds

2003 ◽  
Vol 133 (4) ◽  
pp. 855-873 ◽  
Author(s):  
Seok Woo Kim ◽  
Yong Hah Lee
Keyword(s):  
Riemannian Manifolds ◽  
Schrödinger Operator ◽  
Sobolev Inequality ◽  
Harmonic Functions ◽  
Schrodinger Operator ◽  
Dirichlet Integral ◽  
Local Conditions ◽  
Complete Riemannian Manifolds ◽  
Bounded Harmonic Functions ◽  
Rough Isometry

In this paper, we prove that the dimension of the space of bounded energy-finite solutions for the Schrödinger operator is invariant under rough isometries between complete Riemannian manifolds satisfying the local volume condition, the local Poincaré inequality and the local Sobolev inequality. We also prove that the dimension of the space of bounded harmonic functions with finite Dirichlet integral is invariant under rough isometries between complete Riemannian manifolds satisfying the same local conditions. These results generalize those of Kanai, Grigor'yan, the second author, and Li and Tam.

On the Radial Symmetry Property for Harmonic Functions

2020 ◽  
Vol 64 (10) ◽  
pp. 9-19
Author(s):  
V. V. Volchkov ◽  
Vit. V. Volchkov
Keyword(s):  
Harmonic Functions ◽  
Symmetry Property ◽  
Radial Symmetry
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