The Dirichlet Problem for Harmonic Functions

1980 ◽  
Vol 87 (8) ◽  
pp. 621 ◽  
Author(s):  
Ivan Netuka
2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Anders Karlsson

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.


2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Vladimir Ryazanov

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.


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

2013 ◽  
Vol 7 (1) ◽  
pp. 143-160 ◽  
Author(s):  
Maru Guadie

We study the Dirichlet problem for discrete harmonic functions in unbounded product domains on multidimensional lattices. First we prove some versions of the Phragm?n-Lindel?f theorem and use Fourier series to obtain a discrete analog of the three-line theorem for the gradients of harmonic functions in a strip. Then we derive estimates for the discrete harmonic measure and use elementary spectral inequalities to obtain stability estimates for Dirichlet problem in cylinder domains.


2018 ◽  
Vol 34 (3) ◽  
pp. 1323-1360 ◽  
Author(s):  
Anders Björn ◽  
Jana Björn ◽  
Tomas Sjödin

2013 ◽  
Vol 13 (1) ◽  
Author(s):  
David Hartenstine ◽  
Matthew Rudd

AbstractMotivated by the mean-value property characterizing harmonic functions and recently established asymptotic statistical formulas characterizing p-harmonic functions, we consider the Dirichlet problem for a functional equation involving a convex combination of the mean and median. We show that this problem has a continuous solution when it has both a subsolution and a supersolution. We demonstrate that solutions of these problems approximate p-harmonic functions and discuss connections with related results of Manfredi, Parviainen and Rossi.


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