Tagged Questions
7
votes
2answers
105 views
Mean Value Property of Harmonic Function on a Square
A friend of mine presented me the following problem a couple days ago:
Let $S$ in $\mathbb{R}^2$ be a square and $u$ a continuous harmonic function on the closure of $S$. Show that the average of ...
1
vote
1answer
91 views
Harmonic functions
Let $f: \Omega \to \mathbb{R}$ be a harmonic function, where $\Omega \subset \mathbb{R}^2$ is an open subset. What can be said about the points where $\frac{\partial f}{\partial x} =\frac{\partial ...
1
vote
1answer
82 views
Is this function a subharmonic function?
Does anyone know, is $h\left(z,w\right):=\frac{\left|zw\right|}{\left|z\right|+\left|w\right|}$
for $z$ and $w$ in the unit disk $\mathbb{D}$ of the complex plane a (pluri)subharmonic function? ...
1
vote
0answers
109 views
Notation in Gilbarg/Trudinger? [Section 2.8]
Note: as discussed below, there is no mistake here. The notation and conventions have been cleared up for me. However what I have written is not incorrect, either. At least to me, it is just a ...
1
vote
0answers
155 views
Subharmonic/Superharmonic Inequality in Gilbarg/Trudinger [Section 2.8]
This is in Section 2.8 of Gilbarg and Trudinger. I believe there are some inaccuracies in the proof supplied, and in any case I think there is a more straightforward proof.
Definition:
A ...
2
votes
1answer
76 views
Harmonic function with bounded preimage
I recently saw a question here about bounded/unbounded preimages of a set under a harmonic function. The question asked did not seem to make sense as it was talking about harmonic functions on ...
1
vote
1answer
62 views
Preimage of a point by a non-constant harmonic function on $\mathbb{R}$ is unbounded
Let $u$ be a non-constant harmonic function on $\mathbb{R}$. Show that $u^{-1}(c)$ is unbounded.
I am not getting what theorem or result to apply. Could anyone help me?
