MEAN VALUE PROPERTY OF HARMONIC FUNCTION ON THE HIGHER-DIMENSIONAL SIERPINSKI GASKET

Fractals ◽  
2020 ◽  
Vol 28 (05) ◽  
pp. 2050077
Author(s):  
YIPENG WU ◽  
ZHILONG CHEN ◽  
XIA ZHANG ◽  
XUDONG ZHAO
Keyword(s):  
Harmonic Function ◽  
Harmonic Functions ◽  
Sierpinski Gasket ◽  
Mean Value ◽  
Sierpiński Gasket ◽  
Mean Value Property ◽  
Average Value ◽  
Higher Dimensional ◽  
The Mean

Harmonic functions possess the mean value property, that is, the value of the function at any point is equal to the average value of the function in a domain that contain this point. It is a very attractive problem to look for analogous results in the fractal context. In this paper, we establish a similar results of the mean value property for the harmonic functions on the higher-dimensional Sierpinski gasket.

scholarly journals Some Remarks on Harmonic Functions in Minkowski Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 196
Author(s):  
Songting Yin
Keyword(s):  
Harmonic Function ◽  
Minkowski Space ◽  
Harmonic Functions ◽  
Mean Value ◽  
Mean Value Property ◽  
Minkowski Spaces ◽  
The Mean ◽  
Mean Value Formula

We prove that in Minkowski spaces, a harmonic function does not necessarily satisfy the mean value formula. Conversely, we also show that a function satisfying the mean value formula is not necessarily a harmonic function. Finally, we conclude that in a Minkowski space, if all harmonic functions have the mean value property or any function satisfying the mean value formula must be a harmonic function, then the Minkowski space is Euclidean.

scholarly journals On the mean value property of superharmonic and subharmonic functions

2006 ◽  
Vol 2006 ◽  
pp. 1-3
Author(s):  
Robert Dalmasso
Keyword(s):  
Harmonic Functions ◽  
Mean Value ◽  
Subharmonic Functions ◽  
Mean Value Property ◽  
The Mean

We prove a converse of the mean value property for superharmonic and subharmonic functions. The case of harmonic functions was treated by Epstein and Schiffer.

scholarly journals On the mean value property of fractional harmonic functions

2020 ◽  
Vol 201 ◽  
pp. 112112
Author(s):  
Claudia Bucur ◽  
Serena Dipierro ◽  
Enrico Valdinoci
Keyword(s):  
Harmonic Functions ◽  
Mean Value ◽  
Mean Value Property ◽  
The Mean

On the mean-value property of harmonic functions

1965 ◽  
Vol 14 (1) ◽  
pp. 109-111 ◽  
Author(s):  
Bernard Epstein ◽  
M. M. Schiffer
Keyword(s):  
Harmonic Functions ◽  
Mean Value ◽  
Mean Value Property ◽  
The Mean

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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