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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2answers
510 views
Why are harmonic functions called harmonic functions?
Are they related to harmonic series in any way? Or something else? Wikipedia didn't help.
8
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2answers
329 views
How to argue this consequence?
Suppose that $\Omega=\mathbf{R}^n_+$ and consider a function $0<u<\sup\limits_\Omega u=M<\infty$ such that:
$$\Delta u+u-1=0 \ \ \text{in} \ \ \Omega,$$
$$u=0 \ \ \text{on} \ \ ...
8
votes
0answers
158 views
Kato inequality
For any real-valued smooth function $u$, we have the Kato inequality
$|\nabla|\nabla u||^2\leq(\operatorname{trace}(\operatorname{Hess}(u)))^2$,
which holds when $|\nabla u|\neq0$.
If moreover $u$ ...
7
votes
2answers
105 views
Mean Value Property of Harmonic Function on a Square
A friend of mine presented me the following problem a couple days ago:
Let $S$ in $\mathbb{R}^2$ be a square and $u$ a continuous harmonic function on the closure of $S$. Show that the average of ...
7
votes
3answers
212 views
Bounded, non-constant harmonic functions: how far are they from existing?
Let $f$ be a function that maps $\mathbb{Z}^2$ to $\mathbb{R}$ and consider the operator $T$ which replaces the value of $f$ at $(i,j)$ by the average of the values of $f$ at its four neighbors: $$ ...
6
votes
2answers
137 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.
6
votes
1answer
110 views
Harmonic function.
The function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ given by $f(x) = \|x\|^{2-n}$, where $\|~\|$ denotes the Euclidean norm, is harmonic. This is just a simple computation.
My question is: why ...
5
votes
3answers
650 views
Composition of a harmonic function.
I noticed this question while reading several pdfs of lecture notes, and I'm having trouble figuring it out. Can anyone help?
If $f$ is a harmonic function in a domain $D \subset \mathbb{C}$, and $g$ ...
4
votes
2answers
169 views
harmonic function question
Let $u$ and $v$ be real-valued harmonic functions on $U=\{z:|z|<1\}$. Let $A=\{z\in U:u(z)=v(z)\}$. Suppose $A$ contains a nonempty open set. Prove $A=U$.
Here is what I have so far: Let ...
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 ...
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 ...
4
votes
3answers
117 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 ...
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 $ξ \neq0$ ...
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 ...
3
votes
4answers
191 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 ...
3
votes
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 ...
3
votes
1answer
46 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 ...
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
$$u(re^{i\theta}) = ...
3
votes
1answer
97 views
Dirichlet Problem: Uniqueness of solution
Let $u$ be the solution to a Dirichlet Problem on a bounded open domain $D \subset \Bbb R^n$.
Is the uniqueness of $u$ guaranteed by the maximum principle or by the smoothness of the boundary of $D$?
...
3
votes
1answer
137 views
What is the counter example?
The Liouville's theorem states that if $u$ is a non negative, subharmonic function, $L^\infty(\mathbf{R}^n)$, then $u$ is constant ($n\leq2$). Someone knows a counter example if $n>2$, or where can ...
3
votes
1answer
113 views
Harmonic function with condition on part of its boundary
Suppose $u$ is harmonic in the interior of the unit square $0 \leq x \leq 1$, $0\leq y\leq1$. Suppose furthermore that $u$ and its first derivatives continuously extend to the bottom side $0\leq x ...
3
votes
1answer
72 views
property of harmonic functions
If a real-valued function $u$ is harmonic on a ball $B_{2r}(x)$ in $\mathbb{R}^n$, how would one show that
$\sup_{B_r(x)}u^2\leq\frac{2^n}{|B_{2r}(x)|}\int_{B_{2r}(x)}u^2(y) dy$?
3
votes
0answers
148 views
Superharmonic function and supermartingale
If $f$ is a nonnegative superharmonic function in dimension 2, how to prove that $f$ is constant?
There is an exercise in R.Durrett's probability book, which gives out a method to prove it by ...
2
votes
1answer
51 views
Visualization of subharmonic functions
I have always visualized subharmonic functions as Ahlfors' Complex Analysis thaught me to do: in one dimension lines are harmonic functions and "convex" functions are subharmonic.
I actually just ...
2
votes
2answers
200 views
An inequality about the gradient of a harmonic function
Let $G$ a open and connected set. Consider a function $z=2R^{-\alpha}v-v^2$ with $R$ that will be chosen suitably small, where $v$ is a harmonic function in $G$, and satisfies
$$|x|^\alpha\leqslant ...
2
votes
2answers
71 views
Boundary values of harmonic $u$ are $ u(e^{it}) = 5- 4 \cos t $; find $u(1/2)$ and $v(1/2)$.
My problem is the following:
Let $u$ be a continuous real-valued function in the closure of the unit disk $\mathbb{D}$ that is harmonic in $\mathbb{D}$. Assume that the boundary values of $u$ are ...
2
votes
2answers
112 views
Limit involving the laplacian
I'm trying to prove that if $\Omega$ is an open subset of $\mathbb{R}^n$ and $u$ a $C^2$ function then $$\lim_{r\to 0}\frac{2n}{r^2}\left(u(x)-\frac{1}{|\partial B_r(x)|}\int_{\partial ...
2
votes
1answer
76 views
Removal of singularities for harmonic functions with finite energy
Denote by $B = B(0,1) \subset \mathbb{C}$ the open unit disc and by $B' = B \setminus \{ 0 \} \subset \mathbb{C}$ the punctured unit disc. Assume that $u : B' \rightarrow \mathbb{R}$ is a harmonic ...
2
votes
1answer
119 views
Two question on harmonic function
In a question paper I got the following two questions.
$u(z)=u(x,y)$ be a harmonic function in $\mathbb{C}$ satisfying $u(z)\le a|\ln|z||+b$ for some positive constants $a,b$ and for all complex ...
2
votes
1answer
42 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 ...
2
votes
1answer
148 views
Let $f$ be a harmonic function. Prove that $\overline{f}$ is harmonic.
Let $f$ be a harmonic function. Prove that $\overline{f}$ is harmonic.
I need help to write a rigorous proof. Thank you
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 ...
2
votes
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 ...
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 ...
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\}$$ and its closure with respect to the $L^2$ norm, ...
2
votes
0answers
40 views
Tight bounds for harmonic measure
I recently came across a question concerning harmonic measure here, and was wondering if there is a good reference summarizing different methods of estimating harmonic measure?
Specifically, I would ...
2
votes
0answers
30 views
hyperbolic group ; showing the existence of a ration function with a certain condition
I'm currently working out of a book called differentialgeometry and minimal surfaces written by Jost-Hinrich Eschenburg und Jürgen Jost.
Right now I'm looking at an exercise (12.5) under the ...
2
votes
0answers
326 views
Finding the harmonic conjugate
I have shown that the following function is harmonic and am attempting to find it's harmonic conjugate:
$u=e^{-2xy}\sin(x^2-y^2)$
I know that to find the harmonic conjugate I need to use the ...
1
vote
1answer
53 views
How to determine if the sums and products of harmonic functions is also harmonic?
Suppose I know that $u + iv$ is non-constant and analytic on a domain $D$, then I know that $u$ and $v$ are harmonic on $D$ and not both constant; but how do I then determine whether $3u^2v - v^3 + ...
1
vote
1answer
56 views
Showing particular harmonic function is constant
Suppose $u$ is a real valued continuous function on $\overline{\mathbb D}$, harmonic on ${\mathbb D}$\ $\{0\}$ and $u=0$ on $\partial\mathbb D$, show $\mathbb u$ is constant in $\mathbb D$.
I'm going ...
1
vote
1answer
53 views
Dirichlet problem: Obtaining the harmonic measure through Riesz representation theorem
For the Dirichlet problem on a bounded open domain $D \subset \Bbb R^n$
$$
\Delta u=0, \text{ on } D, \\
\left. u\right|_{\partial D}=f \in C\left( \partial D\right).
$$
With a fix $x$ in $D$, an ...
1
vote
1answer
80 views
is the converse true: in a simply connected domain every harmonic function has its conjugate
The question is.
Is the converse true: In a simply connected domain every harmonic function has its conjugate?
I am not able to get an example to disprove the statement.
1
vote
1answer
372 views
Proving the mean value property of harmonic functions using distributions?
A professor I talked to showed me a proof of the mean value property. (He actually showed it for functions solving the heat equation instead of Laplace's equation, but it seems like the argument is ...
1
vote
1answer
49 views
Nonnegative Superharmonic Function is Constant for $d>2$?
I have to do the following:
Let $\alpha>0$ be fixed, $(X_i)_{i\geq 1}$ be i.i.d., $\mathbb R^{d}$-valued random variables, uniformly distributed on the ball $B(0,a)$. Set $\displaystyle ...
1
vote
1answer
29 views
Harmonic conjugate of $u,v$ in $f=u+iv$
Do I understand correctly the definition of being harmonic conjugate
if I understand it that:
$v$ is the harmonic conjugate of $u$
but
$u$ is not the harmonic conjugate of $v$, but rather $-u$ ?
1
vote
1answer
78 views
How do harmonic function approach boundaries?
Suppose that $D$ is a domain in $\mathbb{R}^n$ (that is, an open, bounded and connected subset), and that $u$ is an harmonic function on $D$. Let $x_0$ be a point at the boundary of $D$.
Question ...
1
vote
1answer
97 views
Existence of solutions for the Dirichlet problem in unbounded domains
Suppose we are trying to solve the Dirichlet problem in a possibly unbounded domain $\Omega \subseteq \mathbb R ^n$ with continuous prescribed boundary data $f$. When $\Omega$ is bounded, it is well ...
1
vote
1answer
79 views
Dirichlet problem: Is the Poisson Integral always a solution?
Let $f$ be continuous on the sufficiently smooth boundary $\partial D$ of a domain $D \subset \Bbb R^n$.
Is the Poisson integral of $f$,
$$
Pf(x)=\int_{\partial D} f(t) ...
1
vote
1answer
255 views
Maximum principle for harmonic functions in unbounded domains
We demonstrated the weak maximum principle for harmonic functions
in bounded domains, proving it first considering the case u
subharmonic, then approximating in this way:
choose $v(x)=x_1^2-M$ so ...
