-4
votes
0answers
29 views

Prove u=0 for a harmonic function u.

Let $\Omega$ be an open connected set in $\mathbb{R}^n$ with the boundary $\partial\Omega$ of class $C^2$. Let $u\in C^2(\Omega)\cap C^1(\overline{\Omega})$ be a harmonic function in $\Omega$, such ...
1
vote
1answer
45 views

Harmonic functions on $\mathbb C-\{0\}$

Find all real-valued $C^2$ differentiable functions $h$ defined on $(0,\infty)$ such that $u(x,y)=h(x^2+y^2)$ is harmonic on $\mathbb C-\{0\}$. This is one of my homework problem. As I understand I ...
0
votes
2answers
46 views

implicit derivates incorperating laplace's equation

If $f(x,y)$ is a harmonic function show that the function $F(x,y)=f(x^2-y^2,2xy)$ is also harmonic. You have to use Laplace's formula to prove this, unless there is an easier way. I'm having trouble ...
1
vote
1answer
40 views

One problem about harmonic functions

Problem. Given open, bounded set $\Omega\subset\mathbb R^d$ with smooth boundary $\partial\Omega$ and given smooth function $\varphi$ on $\partial\Omega$. As known, problem $$ \begin{cases} ...
0
votes
0answers
58 views

Harmonic motion, contract cosinus/sinus expression

I got a harmonic motion given as three partial motions, I am to write the function on the form of: $H(x)= A \cos(\omega(x-x_0)$ The function given is: $H(x) = ...
2
votes
1answer
221 views

Dealing with a non-linear oscillator

This is a problem for my classical mechanics course, but it seems more math, so thats why I am asking this here. So I am given the following equation: $$\ddot x+(x^2+\dot x^2-1)\dot x +x=0$$ $$\dot ...
0
votes
1answer
87 views

Harmonic Conjugate in Star Domain

I have been given that $u(x,y)$ is a harmonic function on a star shaped domain $D$. I have to show that it has harmonic conjugate $v(x,y)$ on same domain given up to additive constant by ...
1
vote
1answer
218 views

Harmonic function in the upper half plane

Find an harmonic function $h(z)$ in the upper half plane with the following properties: $h$ is bounded in $\mathbb{C}_+$ and is continuous in $\{z | \Im(z) \geq 0 \}\setminus \{0 \}$ ...
0
votes
2answers
400 views

Derive the Poisson Formula for a bounded C-harmonic function in the upper half-plane.

My book gives the Poisson Formula for such a harmonic function as: $$ u(x + iy) = \frac{1}{\pi} \int_{-\infty}^{\infty}{\frac{y \cdot u(t) dt}{(t - x)^2 + y^2}} $$ Here is what I have attempted. ...
0
votes
2answers
125 views

Show that a harmonic function $ u(r, \theta) $ dependent only on $ r $ has the form $ u(r, \theta) = a \log r + b $

What I have done so far is this: I've shown that if $ u(r, \theta) $ is a harmonic function dependent only on $ r $ then Laplace's equation becomes $ u_{rr} + \frac{1}{r}u_r = 0 $ I've also shown ...
2
votes
1answer
218 views

Why is the harmonic function $ \log(x^2 + y^2) $ not the real part of any function that is analytic in $ \mathbb{C} - \{0\} $?

I would like to show that $ \log(x^2 + y^2) $ is not the real part of any analytic function in $ \mathbb{C} - \{0\} $ A similar question can be found here, but I don't think this argument is ...
2
votes
1answer
188 views

Showing that if $u$ is a real-valued harmonic function then for any real $c$ we have that $u^{-1}(c)$ is unbounded

I have the following homework question: Let $u$ be a non-constant real-valued harmonic function in $\mathbb{C}$. Prove that $u^{-1}(c)$ is unbounded for every real number $c$ There is a hint ...
1
vote
1answer
144 views

Nonnegative Superharmonic Function is Constant for $d>2$?

I have to do the following: Let $\alpha>0$ be fixed, $(X_i)_{i\geq 1}$ be i.i.d., $\mathbb R^{d}$-valued random variables, uniformly distributed on the ball $B(0,a)$. Set $\displaystyle ...
2
votes
0answers
68 views

Composition of a subharmonic function and a conformal mapping

this is q.4 of p.248 of Ahlfors book: Prove that a subharmonic function remains subharmonic after a composition with a conformal mapping. What I'v tried: Let $u:\Gamma \rightarrow \mathbb{R}$ and ...
0
votes
1answer
202 views

Proving that $f$ is analytic if $f$ and $z f(z)$ are harmonic

If $f$ is harmonic and $zf(z)$ is harmonic, then $f$ is analytic. Please help me prove this. Thanks.
0
votes
0answers
66 views

Find upper and lower bound for $u(3/4)$.

Let $u$ be positive harmonic function in the unit disk such that $u(0)=\alpha$. Find upper and lower bound for number $u(3/4)$. I tried to find an example, that is positive, harmonic( realvalued? ...
2
votes
0answers
67 views

A function in the $L^2$ closure of the set of smooth, harmonic functions on the closed unit disk is smooth and has a harmonic representative.

This is is a homework problem I'm having trouble understanding. I am given the set $$Y=\{u\in \mathbb{C}^{\infty}(\bar{D_1})| \triangle u=0\}$$ and its closure with respect to the $L^2$ norm, ...
1
vote
0answers
66 views

Harmonic Function in $\Omega$ that is continuous in $\overline{\Omega}$ except at a point on the boundary

My problem is the following. Let $x_{0}\in\partial\Omega$ and $\Omega\subseteq\mathbb{R}^{2}$ open and connected domain. Suppose there exists $R\in\mathbb{R}$ such that $\Omega\subseteq B_{x_{0},R}$. ...