scholarly journals Infinite dimension of solutions of the Dirichlet problem

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Vladimir Ryazanov
Keyword(s):  
Dirichlet Problem ◽  
Unit Disk ◽  
Harmonic Functions ◽  
Infinite Dimension ◽  
Boundary Limits ◽  
Nontangential Boundary

AbstractIt is proved that the space of solutions of the Dirichlet problem for the harmonic functions in the unit disk with nontangential boundary limits 0 a.e. has the infinite dimension.

scholarly journals Some remarks concerning harmonic functions on homogeneous graphs

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Anders Karlsson
Keyword(s):  
Dirichlet Problem ◽  
Harmonic Function ◽  
Harmonic Functions ◽  
Cayley Graphs ◽  
Group Cohomology ◽  
Radial Variation ◽  
International Audience

International audience We obtain a new result concerning harmonic functions on infinite Cayley graphs $X$: either every nonconstant harmonic function has infinite radial variation in a certain uniform sense, or there is a nontrivial boundary with hyperbolic properties at infinity of $X$. In the latter case, relying on a theorem of Woess, it follows that the Dirichlet problem is solvable with respect to this boundary. Certain relations to group cohomology are also discussed.

scholarly journals The Dirichlet problem for p-harmonic functions with respect to the Mazurkiewicz boundary, and new capacities

2015 ◽  
Vol 259 (7) ◽  
pp. 3078-3114 ◽  
Author(s):  
Anders Björn ◽  
Jana Björn ◽  
Nageswari Shanmugalingam
Keyword(s):  
Dirichlet Problem ◽  
Harmonic Functions

scholarly journals Several properties of $$\alpha $$ α -harmonic functions in the unit disk

2017 ◽  
Vol 184 (4) ◽  
pp. 627-640 ◽  
Author(s):  
Peijin Li ◽  
Xiantao Wang ◽  
Qianhong Xiao
Keyword(s):  
Unit Disk ◽  
Harmonic Functions
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