Tagged Questions
7
votes
2answers
103 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
24 views
Extracting Harmonic series components
I have a number which is made up of a Harmonic series.
1/2 + 1/3 + 1/4 etc.
Some of the components may not be in the number..
1/2 + 1/7 + 1/11 etc.
Is it possible to recover the individual ...
0
votes
2answers
47 views
Harmonic function, existence of a constant
May i ask you for a little help about a problem with harmonic function? It seems to be not that difficult, in a way even intiutively obvious but i don't really know how to show this explicitly.
We ...
4
votes
1answer
109 views
Simply connected domain and harmonic function
Let $\Omega$ be a simply connected domain that is properly contained in $\mathbb C$, and $u(x,y)$ is harmonic on the unit disk $\mathbb D $, then there is a funtion $f(z)$, that is one-one and ...
4
votes
1answer
158 views
Laplace equation Dirichlet problem on punctured unit ball.
Let $\Omega = \{ x \in \mathbb{R}^n: 0<|x|<1 \}$ and consider the Dirichlet problem
\begin{align}
\Delta u &= 0 \\
u(0) &= 1 \\
u &= 0 ~~~\text{if} ~~|x|=1
\end{align}
By considering ...
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? ...
3
votes
1answer
96 views
Dirichlet Problem: Uniqueness of solution
Let $u$ be the solution to a Dirichlet Problem on a bounded open domain $D \subset \Bbb R^n$.
Is the uniqueness of $u$ guaranteed by the maximum principle or by the smoothness of the boundary of $D$?
...
1
vote
1answer
53 views
Dirichlet problem: Obtaining the harmonic measure through Riesz representation theorem
For the Dirichlet problem on a bounded open domain $D \subset \Bbb R^n$
$$
\Delta u=0, \text{ on } D, \\
\left. u\right|_{\partial D}=f \in C\left( \partial D\right).
$$
With a fix $x$ in $D$, an ...
0
votes
0answers
81 views
Dirichlet Problem: Example where the Green function is not the Poisson kernel
Give an example of a Dirichlet problem where the Green function is not the Poisson kernel.
For a bounded open domain $D$ with a sufficiently smooth boundary and $f \in C\left(\partial D \right)$, the ...
1
vote
1answer
77 views
Dirichlet problem: Is the Poisson Integral always a solution?
Let $f$ be continuous on the sufficiently smooth boundary $\partial D$ of a domain $D \subset \Bbb R^n$.
Is the Poisson integral of $f$,
$$
Pf(x)=\int_{\partial D} f(t) ...
3
votes
1answer
112 views
Harmonic function with condition on part of its boundary
Suppose $u$ is harmonic in the interior of the unit square $0 \leq x \leq 1$, $0\leq y\leq1$. Suppose furthermore that $u$ and its first derivatives continuously extend to the bottom side $0\leq x ...
1
vote
0answers
55 views
Suggestion for a project on Harmonic measure and Fourier analysis
I have a course project on harmonic measure and Fourier analysis.
The goal is to give a presentation on a part of harmonic measure theory which relates to Fourier analysis.
Harmonic measure is a vast ...
0
votes
0answers
55 views
Show that function is a constant
Let $\phi \in L^2(S^{n})$. Let $f=\phi^2$ and let $f_j^m$ be a Fourier coefficients of $f$.
Help me please to show that if
$$
\sum_{j,i}c_jf^m_jY^i_j=\phi,
$$
then $f=constant$.
Here $Y_j^i$ is the ...
