The solutions of the Laplace equation $\Delta f =0$ on a domain $D\subset \mathbb{R}^n$ are known as *harmonic functions*.
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66 views
Harmonic functions in $\mathbb{R}^d$
I want to establish the equivalence of the 3 standard definitions, and that harmonic functions are $C^\infty$. The 3 definitions are:
Mean value property and continuous.
$C^2$ and $0$ Laplacian.
...
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1answer
32 views
Harmonic Function Transformation Help
Consider the harmonic function $$u(x,y)=1-y+\frac{x}{x^2+y^2}$$ on the upper half plane $y>0$.
What is the corresponding harmonic function on the first quadrant $x>0$, $y>0$, under the ...
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1answer
56 views
Showing particular harmonic function is constant
Suppose $u$ is a real valued continuous function on $\overline{\mathbb D}$, harmonic on ${\mathbb D}$\ $\{0\}$ and $u=0$ on $\partial\mathbb D$, show $\mathbb u$ is constant in $\mathbb D$.
I'm going ...
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1answer
39 views
Characterize solutions to Laplace's and Poisson's equation in the unit square with periodic boundary conditions
So I am studying for a qualifying examination and there was this problem from an old exam.
(a) Does the problem $\Delta u = 0$ in the unit square in the plane with u and and all of its partial ...
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1answer
36 views
Corollary to mean value property for harmonic functions?
For $\Omega \subset\mathbb{R}^n$ open, and $u_i:\Omega \to \mathbb{R}$ a sequence of harmonic functions which are uniformly bounded. Prove that for any multi-index $\alpha$ and for any $K \subset ...
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1answer
53 views
How to determine if the sums and products of harmonic functions is also harmonic?
Suppose I know that $u + iv$ is non-constant and analytic on a domain $D$, then I know that $u$ and $v$ are harmonic on $D$ and not both constant; but how do I then determine whether $3u^2v - v^3 + ...
2
votes
1answer
46 views
Mean Value Property
I'm currently studying the theory of PDEs and, in particular, harmonic functions.
I've been given this question:
Show that if $u:(a,b) \rightarrow \mathbb{R}$ is continuous, and satisfies the ...
2
votes
1answer
19 views
The extension of smooth function under the restriction of its Laplacian
$u$ is a smooth bounded function on $\Omega-\{0\}$ where $\Omega$ is an open neighborhood of $0$ in $\mathbb R^n$. If $\Delta u$ is a bounded function on $\Omega-\{0\}$, then can we extend $u$ to be a ...
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0answers
30 views
Angle between Harmonic functions
I have a question about a practice prelim problem:
What is the angle between the curves $Re(z^3) = 1$ and $Re(z^3) = Im(z^3)$?
Also, What is the angle between the curves $Re(z^3) = 0$ and $Re(z^3) ...
7
votes
2answers
106 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 ...
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0answers
33 views
Prove that the function is bounded and harmonic
Let $ f: \{ z: $ Re $z$ = $0$ $\}$ $\rightarrow$ $\mathbb {R}$ be a bounded continuous function and define $ u: \{ z: $ Re $z$ > $0$ $\}$ $\rightarrow$ $\mathbb {R}$ by
$$
u(x+iy) = \frac{1}{\pi} ...
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1answer
24 views
Extracting Harmonic series components
I have a number which is made up of a Harmonic series.
1/2 + 1/3 + 1/4 etc.
Some of the components may not be in the number..
1/2 + 1/7 + 1/11 etc.
Is it possible to recover the individual ...
1
vote
1answer
50 views
Nonnegative Superharmonic Function is Constant for $d>2$?
I have to do the following:
Let $\alpha>0$ be fixed, $(X_i)_{i\geq 1}$ be i.i.d., $\mathbb R^{d}$-valued random variables, uniformly distributed on the ball $B(0,a)$. Set $\displaystyle ...
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1answer
31 views
Harmonic functions and real valued function related to it
Find all real-valued functions $h$, defined and of class $C^2$ on the positive real line, such that the function $u(x,y)=h(x^2+y^2)$ is harmonic.
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0answers
38 views
Bounded harmonic functions on the right half plane
Let $ f: \{ z: $ Re $z$ = $0$ $\}$ $\rightarrow$ $\mathbb {R}$ be a bounded continuous function and define $ u: \{ z: $ Re $z$ > $0$ $\}$ $\rightarrow$ $\mathbb {R}$ by
$$
u(x+iy) = \frac{1}{\pi} ...
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0answers
35 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 ...
3
votes
4answers
196 views
A question in Complex Analysis $\int_0^{2\pi}\log(1-2r\cos x +r^2)\,dx$
My problem is to integrate this expression:
$$\int_0^{2\pi}\log(1-2r\cos x +r^2)dx.$$
where $r$ is any constant in $[0,1]$.
I know the answer is zero. Can you explain you idea to me or just prove ...
2
votes
1answer
53 views
Visualization of subharmonic functions
I have always visualized subharmonic functions as Ahlfors' Complex Analysis thaught me to do: in one dimension lines are harmonic functions and "convex" functions are subharmonic.
I actually just ...
0
votes
1answer
19 views
Weighted averages in harmonic functions.
Is it the case that for a harmonic function on a graph any value of the interior point is the weighted average of the boundary points? I know that for a harmonic function each point is the weighted ...
3
votes
1answer
47 views
Bound for analytic function from unit disk into punctured unit disk
Suppose $f$ is analytic in the unit disk $D$ and satisfies $0<|f(z)|<1$. Show that $|f(z)|\leq|f(0)|^{\frac{1-|z|}{1+|z|}}$ for all $z\in D$.
I tried to work with $\log|f|$. It seems that ...
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0answers
27 views
Question about how to use some functionals and calculate some harmonics
I need to apply some transforms to some images using some functionals applied to a set of values extracted from the pixels of the images. The problem is I don't know if I'm understanding the ...
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votes
1answer
112 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.
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1answer
71 views
Poisson integral
Given a bounded harmonic function $u(z)$ on the open unit disk, and given radial limits $\lim_{r\rightarrow 1^{-1}}u(re^{i\theta})$ being some constant $a$ for $0<\theta<\pi$ and being some ...
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votes
1answer
27 views
Conformal map projecting a line to a sine wave
I'm looking for an analytic complex function that will map a straight line on to a sine wave. Are there any known examples?
To be more specific, let
$f(x+iy) = u(x,y) + iv(x,y)$.
I want to find a ...
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0answers
35 views
Geometric Condition for Harmonic Function
Under what geometric condition on real harmonic functions u and v on a region G is the function uv also harmonic?
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2answers
42 views
example of sub-harmonic function
A continuous function $u:\mathbb{R}^n\to\mathbb{R}$ is sub-harmonic if for every $x_0\in\mathbb{R}^n$ and $r>0$
$$u(x_0) \leq \frac{1}{|\partial B(x_0,r)|}\int_{\partial B(x_0,r)} \!\!u(x)\ ...
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1answer
29 views
Harmonic conjugate of $u,v$ in $f=u+iv$
Do I understand correctly the definition of being harmonic conjugate
if I understand it that:
$v$ is the harmonic conjugate of $u$
but
$u$ is not the harmonic conjugate of $v$, but rather $-u$ ?
2
votes
0answers
30 views
For compact $K \subset \mathbb C$, show that $u(z) = -\log(\mathrm{dist}(z,K))$ is subharmonic
Let $K \subset \mathbb C$ be compact, and let $u(z) = -\log(\mathrm{dist}(z,K))$ be defined on $\mathbb C \setminus K$. May I get a hint for proving that $u(z)$ is subharmonic?
Subharmonic, here, is ...
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votes
2answers
49 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 ...
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votes
0answers
46 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? ...
4
votes
3answers
118 views
Is it a harmonic function or not?
I am trying to resolve a question whether a certain function is harmonic or not. If yes, I should find its harmonic conjugate.
The function is $u = \frac{x}{x^2+y^2}$.
I found that it is a harmonic ...
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vote
0answers
126 views
Green's function for the Laplace operator $\Delta$ in a rectangle (or square)?
What I'm looking for is an explicit form of the Green's function $G$ for the Laplace operator $\Delta$ in a rectangular (or square) domain $D\subset\mathbb R^2$, e.g. $D=[0,1]\times[0,1]$. Namely, I'm ...
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vote
0answers
93 views
Biharmonic operator
Consider the problem:
$$ \Delta^2 u = f$$
on the square domain $U=(0,1)\times(0,1)$ with boundary conditions:
$$ u(x,y)=\Delta u(x,y) = 0$$
for $(x,y) \in \partial U.$
I try to solve it with the ...
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vote
1answer
141 views
Derive Poisson's integral formula for Im z>0
How to derive Poisson's integral formula for $\text{Im }{z}>0$ given that for $|z|<1 $ we have ...
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votes
2answers
139 views
How do you prove that $\ln|f(z)|$ is harmonic?
Suppose that $f(z)$ is analytic and nonzero in a domain $D$. Prove that $\ln|f(z)|$ is harmonic in $D$.
I know the laplacian equation but I'm not sure how to use it.
2
votes
0answers
49 views
A function in the $L^2$ closure of the set of smooth, harmonic functions on the closed unit disk is smooth and has a harmonic representative.
This is is a homework problem I'm having trouble understanding. I am given the set $$Y=\{u\in \mathbb{C}^{\infty}(\bar{D_1})| \triangle u=0\}$$ and its closure with respect to the $L^2$ norm, ...
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votes
0answers
52 views
Elementary question about differentiating harmonic complex functions
Consider the complex analytic function $\Omega(z) = \phi(x,y,) + \psi(x,y)$ in a domain D. If we transform from z to w using $w=f(z), w = u+ iv$, with f(z) analytic in D, with the co-domain in the w ...
4
votes
1answer
106 views
Harmonic functions with zeros on two lines
For which pairs of lines $L_1$, $L_2$ do there exist real functions, harmonic in the whole plane, that are $0$ at all points of $L_1 \cup L_2$ without vanishing identically?
Note: This is ...
3
votes
1answer
107 views
Characterization of positive harmonic functions on unit disc with $0$ radial limits
Suppose $u$ is a positive harmonic function in $U$, and $u(re^{i\theta}) \to 0$ as $r \to 1$, for every $e^{i\theta} \ne 1$. Prove that there is a constant $c$ such that
$$u(re^{i\theta}) = ...
4
votes
2answers
124 views
Fourier transform of $|x|^{-t}$
In $\mathbb{R}^d$, let $f(x)=|x|^{-t}$, its Fourier tranform $F(f)(ξ):=(2\pi)^{-\frac{d}{2}}∫_{\mathbb{R}^d} e^{ix\cdot ξ}f(x)dx$, is there any fast way to see that this integral converge at $ξ \neq0$ ...
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votes
1answer
58 views
if $\Delta u \geq c$ for some $c>0$ then $u$ has a max on the boundary
Let $D=\{(x,y): \vert(x,y)\vert \leq 1\}$ and let $u:D\rightarrow \mathbb R$ be continuous function with three continous derivatives in the interior of $D$.
Show that if there is a number $c>0$ ...
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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 ...
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
159 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
91 views
Discontinuity of double-layer potentials
I'm currently reading about solutions to boundary-value problems for Laplace's equation, and I'm a bit confused with regards to the discontinuity properties of double-layer potentials. So the text ...
0
votes
1answer
90 views
Harmonic Extension
Let be $u$ a harmonic function defined on an open set $\Omega \setminus \{p\} \subset \mathbb{C}$ of the complex plane. Show that if $u$ is bounded in a neighborhood of $p$ then $u$ admits a harmonic ...
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1answer
104 views
Questions about harmonic functions and distribution.
If harmonic functions converges in the distribution sense to a distribution. Then can we prove that the functions are actually converges uniformly to a function on every compact set. And the limit ...
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1answer
59 views
Harmonic Polynomial Function
I can't figure out this question:
For what values of the constants A and B is the polynomial function
$F(x,y) = (-5)x^5 + Ax^{3}y^{2} + Bxy^{4}$
harmonic in the whole $xy$-plane?
$A=?$
$B=?$
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1answer
66 views
Harmonic function product, Knowing that one is Harmonic implies something about the other?
Hello I have been fighting with this problems for a couples of days and cant get anything better, I will thanks a lot for your help, for little hint, some clue or idea :
Let $A$ be an Harmonic ...
1
vote
1answer
78 views
How do harmonic function approach boundaries?
Suppose that $D$ is a domain in $\mathbb{R}^n$ (that is, an open, bounded and connected subset), and that $u$ is an harmonic function on $D$. Let $x_0$ be a point at the boundary of $D$.
Question ...
