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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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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59 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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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 ...
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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 ...
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114 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 ...
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42 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 ...
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130 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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213 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
377 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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73 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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49 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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39 views

How to show by example that existence of barrier function of any set $U\subset \mathbb{C}$ is dependent of its set?

How to show that there a set that has no barrier function? I mean that how to show by example that existence of barrier function of any set $U\subset \mathbb{C}$ is dependent of its set. Definition ...
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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
60 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 $-u$ ?
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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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104 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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67 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? ...
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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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482 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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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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310 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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660 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.
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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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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 ...
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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}) = ...
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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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Weakly Harmonic Functions (Weak Solutions to Laplace's Equation $\Delta u=0$) and Logic of Test Function Techniques.

In analysis we often use test functions $\phi\in C_{0}^{\infty}(U)$ in order to make some kind of deduction about another function $u:U\mapsto\mathbb{C}$. For example, if one can obtain the ...
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94 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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151 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 ...
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209 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 ...
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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 ...
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211 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 ...
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245 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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249 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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77 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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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 ...
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141 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 ...
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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 $\Omega\subseteq B_{x_{0},R}$. ...
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147 views

How to prove that this is an Harmonic funtion?

Let $u$ be an Harmonic function in $B(0,a)$ in $R^3$ we define $I(x)=x\dfrac{a^2}{|x|^2} $ Let $w(x) = u(I(x))$. Is there a way to prove that $w$ is harmonic without making too much computation? ...
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1answer
226 views

Maximum of strictly subharmonic function

Let $u\in C^2(D)$, $D$ is the closed unit disk in $\mathbf{R}^2$. Assume that $\Delta u>0$. Show that $u$ cannot have a maximum point in $D\setminus\partial D$. This statement is in a calculus ...
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104 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? ...
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Elliptic equations and harmonic functions

Hi, I need some help with the following problem: Let $u(x_0,y_0)$ be a point of the boundary of a domain $\Omega$ contained in a circle of radius $R$ with center at $(x_0,y_0)$. Let $u$ be an ...
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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 $D$? ...
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270 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 ...
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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 ...
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2answers
260 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 ...
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167 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 ...
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82 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 ...
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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} \ \ ...
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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) ...