The Dirichlet Problem for Relative Harmonic Functions

1984 ◽  
pp. 98-140
Author(s):  
J. L. Doob
Keyword(s):  
Dirichlet Problem ◽  
Harmonic Functions

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 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.

The Dirichlet Problem for Harmonic Functions

1980 ◽  
Vol 87 (8) ◽  
pp. 621 ◽  
Author(s):  
Ivan Netuka
Keyword(s):  
Dirichlet Problem ◽  
Harmonic Functions

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 Stability estimates for discrete harmonic functions on product domains

2013 ◽  
Vol 7 (1) ◽  
pp. 143-160 ◽  
Author(s):  
Maru Guadie
Keyword(s):  
Fourier Series ◽  
Dirichlet Problem ◽  
Harmonic Measure ◽  
Harmonic Functions ◽  
Discrete Analog ◽  
Stability Estimates ◽  
Product Domains ◽  
Discrete Harmonic Functions

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.

scholarly journals The Dirichlet problem for $p$-harmonic functions with respect to arbitrary compactifications

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

Statistical Functional Equations and p-Harmonious Functions

2013 ◽  
Vol 13 (1) ◽  
Author(s):  
David Hartenstine ◽  
Matthew Rudd
Keyword(s):  
Functional Equation ◽  
Dirichlet Problem ◽  
Harmonic Functions ◽  
Functional Equations ◽  
Continuous Solution ◽  
Convex Combination ◽  
Mean Value ◽  
Mean Value Property ◽  
The Mean ◽  
Statistical Functional

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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