# Tagged Questions

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 ...
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 ...
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 ...
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 ...
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 ...
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? ...
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 $D$? ...
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 ...
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 ...
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) ...
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 ...
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 ...
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 ...
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