For questions regarding harmonic functions.

learn more… | top users | synonyms (2)

1
vote
2answers
169 views

Is there a conformal mapping that sends the upper semi-circle to the positive (or negative) real line,

and the real interval [-1,1] to the other half of the real line? I am considering the upper semi-circle ${|z|=1, 0<arg(z)<\pi}$, with the line [-1,1] that closes the loop. I want to map the ...
1
vote
1answer
30 views

Showing an equation satisfies laplace equation

I'm completely lost with the laplace equation I've searched different explanations of it on google and on this website and nothing is helping explain it. The question I was given is: Show that the ...
1
vote
1answer
104 views

Using Green's identity to show that a harmonic function with zero boundary values is identically zero

I am confused how to do this question. I need to use Green's first identity and if $\nabla(f)=0$ then $f$ is constant on $\Omega$ since $\Omega$ is path connected. I have subbed in the information ...
0
votes
0answers
44 views

How to find a family of analytic functions with a modulus of boundary value = 1, and no zeros inside the unit disk?

a) Find all $f(z)$ satisfying i) is analytic in $|z|\le 1$ ii) has no zeros in $|z|<1$ iii) $|f(e^{i\theta})|=1$ for $0<\theta<2\pi$ b) What are the possible $f(z)$ if (ii) is replaced by ...
0
votes
1answer
53 views

Finding a harmonic function on half-disk that is equal to $1$ on the semi-circle and $0$ on the diameter

I first showed that the mapping $$z + \frac{1}{z}$$ sends the upper semi-disk, $\{|z|<1, \operatorname{Im} z >0\}$, along with the real line from $-1$ to $1$, to the whole of the real line in ...
1
vote
0answers
17 views

Laplace's equation in cylindrical coordinates for a function that only depends on the angle

I need to solve Laplace's equation: $$\nabla^2\Phi = 0$$ with the boundary conditions: $$\Phi(\theta=0)=0$$ $$\Phi(\theta=\pi)=a_1$$ In cylindrical coordinates ($r,\theta,z$), for $\Phi ...
1
vote
1answer
14 views

Harmonic measure function and the surjectivity of the diagonal map

Let $\Omega$ be a finitely connected bounded domain in the complex plane bounded by $n+1$ analytic jordan curves. Letting $\partial \Omega$ denote the boundary of $\Omega$, we write $\partial \Omega = ...
0
votes
1answer
51 views

Solving Laplace's equation for semi-infinite bar using conformal mapping

I am trying to solve the 2d version of Laplace's equation ($\frac{\partial^2 V}{\partial x^2}+\frac{\partial^2 V}{\partial y^2}=0$) for the semi-infinite bar shown below in the shaded region. That ...
2
votes
2answers
37 views

How to prove this function is bounded?

I've got the following function: $v(r,\theta) = \frac{1}{2\pi} \int_{0}^{2\pi} \frac{r^{2} - 1}{r^{2} + 1 - 2 r \cos\left(\psi - \theta)\right)}f(\psi)d\psi$ The function $f$ is NOT specified, but ...
0
votes
0answers
45 views

valid kernel and invertibility

Perhaps it is an easy question and I should be able to do it. However I have no idea how to work with it. I am trying to show that following kernal is valid: K(x,y) = 1/(1-xy) where x and y are ...
2
votes
0answers
32 views

Helmoltz equation on the torus

I am looking for the solution of the Helmoltz equation (or even Laplace, if not available) on the torus, that is, the manifold of line element \begin{equation} ds^2 = (c + a \sin(\theta))^2 ...
0
votes
0answers
50 views

Convex function on $\mathbb{R}^n$ composed with $n$ subharmonic functions

Currently, I am working on the following question (which appears in Hörmander's Notions of Convexity, Exercise 3.2.6) Suppose $f:\mathbb{R}^n\to\mathbb{R}$ is a convex function which is increasing ...
0
votes
2answers
63 views

Harmonic function and Neumann Compatibility Condition

______________________________________________________________ Conversely, in a different approach using Green's 1st identity, I showed that by choosing v ≡ 1, that the compatibility condition is ...
1
vote
2answers
51 views

Show that $u(x,y)$ is constant.

Let $u(x,y)$ be a harmonic function on domain s.t all the partial derivatives of $u(x,y)$ vanish at the same point of , then $u(x,y)$ is constant. Now the thing is if the harmonic conjugate of ...
0
votes
0answers
31 views

Prove Poisson kernel of the upper half-plane is the Cauchy distribution

Let $d=2$, and consider the domain $D=H$, the upper half-plane. Show that the Poisson kernel is the Cauchy distribution by following the steps below. (A) Let $W_t=(X_t,Y_t)$. Show that for any ...
0
votes
1answer
54 views

Laplace equation problem: difficulty finding a particular solution given the boundary conditions.

Solve Laplace's equation: $$\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0$$ for the region $\{(x, y) : 0 < x < 1, 0 < y < 1\}$, given these boundary conditions: ...
0
votes
1answer
36 views

When finding a harmonic function, from a conformal mapping, is it ok to have imaginary numbers in the argument,

Say after finding a conformal mapping, using this conformal mapping, I find a harmonic function $$\arg \left[\frac{(z^2+1)}{(z^2-1)}\right]$$ that satisfies boundary conditions that I want. Is it ...
1
vote
1answer
34 views

How can I find the value of…

Please help me finding the value of the following integral. If $U\subseteq \mathbb{C}$ is an open set and $z_0\in U$ and $r>0$ and $\{z:|z-z_0|\le r\}\subseteq U$ and $j\in \mathbb{Z}^+$ $$\large ...
0
votes
1answer
32 views

Having trouble using my conformal mapping to produce a harmonic function on the quarter disk,

I'm currently working on a problem that asks to map the unit quarter disk from quadrant I to the upper half-plane, and then to use this mapping to find a harmonic function on the quarter disk, taking ...
0
votes
0answers
19 views

General multipole solution to Laplace equation in polar coordinates

I am seeking the general solution for the Laplace equation in cylindrical coordinates or $\nabla^2 \omega = 0$. In several texts, the general solution can be found via separation of variables and ...
0
votes
1answer
59 views

Why real valued harmonic functions are holomorphic.

Let $f$ be a real valued harmonic function on $C,$ then Claim $g= \frac {\partial f}{\partial x} - i \frac {\partial f}{\partial y} $ as holomorphic and $h= \frac {\partial f}{\partial x} + i \frac ...
0
votes
1answer
47 views

Why is a harmonic conjugate unique up to adding a constant?

If $v$ and $v_0$ are harmonic conjugates of $u$, then $u + iv$ and $u + iv_0$ are analytic functions. Then $i(v - v_0)$ is analytic, but how does this imply $v - v_0$ is a constant function?
0
votes
1answer
70 views

Laplace's equation in an infinite strip - bounded vs. unbounded solution

Let $D = \{\ ( x,y )\in \mathbb{R}^{2} \ | \ 0<y<1\ \}$. Let $A,B\in\mathbb{R}$. Consider the boundary value problem: $\Delta u = 0$ $u(x,0)=A\ $ and $\ u(x,1)=B\ $ for all $x \in \mathbb{R}$ ...
1
vote
1answer
39 views

If $\Delta u - u \geq 0$ in $D$, show that $u\leq \max\{\ \max\limits_{\partial D}(u), \ 0 \ \}$

Let $D$ be a bounded domain, with boundary $\partial D$. Suppose $u \in C^{2}(D)\cap C^{0}(D\cup\partial D)$ satisfies $\Delta u - u \geq 0$ in all of $D$. I am supposed to prove then that $u\leq ...
1
vote
1answer
95 views

Maximum principle for subharmonic functions

Let $\Omega$ be a domain of $\mathbb{R}^n$, and $u:\Omega\to\mathbb{R}$ a continuous function. We call $u$ subharmonic if for any ball $B\subset\subset\Omega$ and any $h:\overline B\to\mathbb{R}$ ...
1
vote
1answer
31 views

Trying to compute an integral using Dirichlet's problem solution

I want to compute for $r < 1$ that $$ r \cos \phi = \frac{1}{2 \pi} \int_0^{2 \pi} \frac{ (1-r^2) \cos \theta }{1 - 2r \cos( \phi - \theta) + r^2 } d \theta $$ In my notes, it says that the way ...
0
votes
0answers
26 views

Harmonic functions on bounded regions: constant on boundary

Suppose $u$ is a harmonic function on the bounded region $B(\mathbf{0},1) \subset \mathbb{R}^2$. Suppose I am given the boundary condition $u = c$ (where $c\in\mathbb{R}$) on $\partial ...
2
votes
0answers
25 views

Net flux zero equivalent to vanishing solution?

Consider \begin{align*} \Delta u(P) = 0, \quad P\in \Omega,\\ \frac{\partial u(P)}{\partial n_{p}} = f(P),\quad P\in \Gamma, \end{align*} where $\Omega$ is the infinite domain in the plane outside ...
0
votes
1answer
39 views

Proove that $x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y}$ is the real part of an analytic function.

The problem is as follows: Proove that $U(x,y) = x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y}$ is the real part of an analytic function. where $f(z)$ is analytic such that ...
2
votes
1answer
60 views

Have I found my harmonic function correctly?

The problem statement is: Use conformal mapping to find a harmonic function $U(z)$ defined on the unit disc $∣z∣<1$ such that $$\lim_{r→1} U(re^{iθ}) = \left\{\begin{aligned} &+1 &&: ...
3
votes
2answers
43 views

Underdamped free vibration proof

I need to prove the solution form of: $$y''+2cwy'+wy=0$$ My book says, after assuming a solution of the form $Ce^pt$, you can show that: $$y=[A\sin(wt)+B\sin(sw)] \cdot e^pt$$ I tried using the ...
0
votes
0answers
39 views

Subharmonic function.

Does anyone know where I can find properties involving integral average for a function that is subharmonic for heat equation, i.e., $$ u_t-\Delta u\leq0. $$ I need something like $$ u(x,t)\geq ...
0
votes
1answer
14 views

Helmholtz solutions on compact domains

Consider two compact domains $\Omega_1$ and $\Omega_2$ in $\mathbb{R}^n$, such that $\partial \Omega_1 \cap \partial\Omega_2$ is a real analytic hypersurface. Suppose I have an eigenfunction $\varphi$ ...
0
votes
1answer
55 views

Cylindrical Harmonics - Can't Find Bessel Equation

I'm solving Laplace's equation in a region with cylindrical symmetry (i.e. no polar angle dependence). Thus, from the outset, Laplace's equation becomes, $$ ...
0
votes
1answer
44 views

Maximum Modulus principle on Rudin's theorem.

I have studied rudin's Real and complex analysis, and have a question on the proof that why the level set $E$ is compact. Could you give me some hint? 11.13 Theorem If a continuous function $u$ ...
2
votes
1answer
35 views

Is subharmonic property preserved under mean value integral

In reading a paper, I've come across an "obvious" statment that looks funky. To make the question applicable to a wider audience let me simplify it a bit: Suppose $h$ is subharmonic on the ...
0
votes
0answers
33 views

External and internal multipole expansion for axisymmetric potential - the region of convergence.

Say, we have a system of electrodes exhibiting symmetry around a certain axis. We know the explicit expression for the potential on the axis $\phi (z)$. We want to find the potential for any point in ...
1
vote
1answer
16 views

Smooth extensions from boundary of convex domains

Given a convex domain $\Omega$ with piecewise $C^1$ boundary data $g \in C(\partial \Omega)$ and $g \in C^1(\Gamma_i)$ with $\bigcup_{i} \Gamma_i = \partial \Omega$. Now, I want to know if there ...
1
vote
0answers
47 views

PDE: Laplace equation Maximum Principle

The maximum priciple for Laplace equation assumes, in both PDE textbooks by Fritz John and Lawrence Evans, that the domain of the harmonic solution be bounded. Is the maximum priciple still valid if ...
1
vote
0answers
62 views

Fourier transform on Laplace equations

We can solve some Laplace equations on the lectangular area by using Fourier transform. However many textbooks only introduce the cases when the area is given as a half plane or a strip. ; $y>0, ...
0
votes
0answers
27 views

Harmonic conjugates

Suppose $u$ is harmonic. We know by definition of chain rule that $$dv = \frac{\partial v}{\partial x} dx+\frac{\partial v}{\partial y} dy.$$ There is also a known formula of $$dv=-\frac{\partial ...
3
votes
1answer
42 views

Bounding harmonic functions.

Assume $f, g: \mathbb{R}^d \to \mathbb{R}$ are harmonic functions. Assume that there exist $C < \infty$ and $\alpha < 1$ such that for all $x$,$$|f(x)| \le C|x|^\alpha.$$What is the easiest ...
0
votes
0answers
61 views

Poisson Integral Formula for Upper Half Plane

It is known that the Poisson Integral Formula: $f(x,y) = \dfrac{y}{\pi}\displaystyle\int\limits_{\mathbb{R}}\dfrac{U(t)}{(x-t)^{2}+y^{2}}dt$ serves as a solution to the Laplace equation on the upper ...
1
vote
0answers
16 views

Weighted Analogue of Mean Value Property

Let $f$ be a harmonic function defined on a planar disk $B \subset \mathbb{R}^2$ and continuous on $\partial B$. For which probability measures $\mu$ on $\partial B$ is it true that $$\int_{\partial B ...
0
votes
1answer
33 views

Bounded solution to $\Delta u = f$ with certain boundary conditions tend to 0?

Let $T := \{(x, y) \in \mathbb{R}^{2}: x \geq 0, y \geq 0\}$ and suppose $f$ is a continuous function which vanishes when $x^{2} + y^{2} > R$ for some $R$. Suppose $u$ solves $$\Delta u = f \text{ ...
0
votes
1answer
49 views

Properties of a function of a harmonic function

Suppose that u : $\mathbb{R} \rightarrow \mathbb{R}$ is such that, whenever v : $\mathbb{R}^n \rightarrow \mathbb{R}$ is harmonic, so is u(v(x)). What can you say about u? My first inclination is ...
1
vote
0answers
30 views

Numerical evaluation of the Laplace operator near the singularity points in spherical coordinates

If one had to evaluate $\Delta Y_l^m$ numerically everywhere on the unit sphere, including the singularity points $\theta = 0,\pi$, how would they do it? Let's say $Y_l^m$ is a spherical harmonic. I'm ...
0
votes
1answer
23 views

Show that the imaginary part of $\frac{z^2}{z-z_p}$ is harmonic

Let $z\in\Omega \subset\mathbb{C}$ and $z_p\notin \Omega$. Show that $\text{Im}(\frac{z^2}{z-z_p})$ is harmonic in $\Omega$, where $\text{Im}(z)$ is the imaginary part of $z$. So far: For $z = \alpha ...
2
votes
1answer
25 views

Let $u(x)$ be harmonic for $x \in \Bbb R^3$. Suppose that $\nabla u \in L^2 (\Bbb R^3)$. Prove that $u$ is a constant.

Let $u(x)$ be harmonic for $x \in \Bbb R^3$. Suppose that $\nabla u \in L^2 (\Bbb R^3)$. Prove that $u$ is a constant. I tried with Green's equation but it didn't work. Could anyone tell me which ...
1
vote
1answer
90 views

Reflection from the upper half ball to the whole ball is harmonic

I have a question about problem 9(b) in Chapter 2 of Evans' PDE book. It says if we have $u$ is harmonic in the open upper half ball $U^+$ and $u\in C^2(U^+)\cap C(\bar{U^+})$, $u=0$ for $x\in ...