The solutions of the Laplace equation $\Delta f =0$ on a domain $D\subset \mathbb{R}^n$ are known as *harmonic functions*.

learn more… | top users | synonyms (2)

6
votes
1answer
251 views

Describing co-ordinate systems in 3D for which Laplace's equation is separable

Laplace's Equation in 3 dimensions is given by $$\nabla^2f=\frac{ \partial^2f}{\partial x^2}+\frac{ \partial^2f}{\partial z^2}+\frac{ \partial^2f}{\partial y^2}=0$$ and is a very important PDE in ...
2
votes
1answer
24 views

introduction to potential theory in $\mathbb{R}^3$ [on hold]

A differentiable function $g: \mathbb{R}^3 \to \mathbb{R}$ is said to be harmonic in a subset $B \subset \mathbb{R}^3$ if $\Delta^2 g = 0$ for all $p \in B$. Let $M \subset \mathbb{R}^3$ be a bounded ...
0
votes
0answers
39 views

Removable singularity of a harmonic function

Assume that $h$ is harmonic in the punctured unit disk $\mathbb D\backslash\{0\}$ such that $$ \lim_{r\to0}h(re^{it})=0 $$ for all $t\in\mathbb R$. Can $h$ be extended to a function harmonic in ...
3
votes
1answer
299 views

How to solve the two dimensional Laplace's equation for certain cases?

Had a doubt regarding Laplace's equation. In many textbooks, the general solution to the two dimensional Laplace equation is mentioned as: $$\Phi(\rho,\phi) = A_{0} + B_{0}\ln(\rho) + ...
4
votes
0answers
19 views

Harmonic functions and null recurrence

Let $(X_n)_{n \geq 0}$ be a irreducible Markov chain defined on a countable state space $S$. It is known some ways to figure out if this chain is recurrent or not looking for superharmonic functions ...
0
votes
1answer
19 views

Calculate $\int_{B_R(0)}u$ for a very weak solution $u$ of a Dirichlet boundary problem

Let $B_R:=B_R(0)\subseteq\mathbb{R}^n$ be the open ball with radius $R>0$ around $0$, $f\in\mathcal{L}^\infty(B_R)$ be radial and $u$ be a very weak solution² of $$\left\{\begin{matrix}-\Delta ...
0
votes
1answer
16 views

Meanvalue formula of harmonic functions - translation invariance of surface measure

In almost every proof of the mean value formula for harmonic functions (e.g. Evans 2.Theorem 2) one finds the following calculation $$ \frac{1}{|{\partial B(0,1)}|}\int_{\partial B(0,1)} D ...
1
vote
1answer
22 views

Show that a harmonic function on an open connected set which is holomorphic on some open subset is in fact holomorphic everywhere.

Suppose $f$ is a harmonic function on a connected open set $\Omega$ in the complex plane, and suppose also that $f$ is holomorphic on some open subset $U$ of $\Omega$. Prove that $f$ is holomorphic on ...
0
votes
1answer
28 views

Show that if $g$ is nonconstant holomorphic and $f$ is harmonic such that $fg$ is harmonic, then $f$ is holomorphic.

Let $\Omega$ be an open and connected set in the complex plane and $g$ be a nonconstant holomorphic function on $\Omega$. Show that if $f$ is harmonic on $\Omega$ such that $fg$ is also harmonic on ...
2
votes
1answer
51 views

3-dim Brownian motion, harmonic function and its expectation

Given $f(x)=\frac{1}{|x+z|}$, a function from $\mathbb{R}^3\backslash \{z\}$ to $\mathbb{R}$, $z \in \mathbb{R}^3\backslash \{0\}$ and $B$ a 3-dim Brownian motion. I had succes showing that this ...
2
votes
1answer
35 views

Using conformal mapping to solve a boundary value problem,

Use conformal mapping to solve the following boundary value problem for $u=u(x,y)$ in the planar region $R=\{(x,y) \in \mathbb{R}^2: x^2 + y^2 > 1 \text{ and } y>0\}$: u solves ...
1
vote
0answers
19 views

Harmonic function on a wedge

Find a harmonic function $\phi(r,\theta)$ in the wedge with three sides $\theta=0$, $\theta=\beta$ and $r=a$ and boundary conditions $\phi(r,0)=0=\phi(r,\beta)$, $0<r<a$ and ...
3
votes
0answers
82 views

Prove or disprove: If $h$ is harmonic on $E$, then $h$ is constant on each $C_i$

For a general finite Markov chain $(X_n)_{n\in\mathbb{N}_0}$ with state space $E$ and transition matrix $P=(p_{x,y})_{x,y\in E}$, not necessarily irreducible, we define the linear space of ...
0
votes
1answer
21 views

Harmonic / Analytic functions

Show that if φ(x, y) is harmonic in a domain D, then f(z) = φ$\\_{x}$(x, y) − iφ$\\_{y}$(x, y) is analytic in D. I figure that φ$\\_{xx}$ = - φ$\\_{yy}$ from "φ(x, y) is harmonic" so then that means ...
0
votes
0answers
36 views

How can I mathematically proof an incoherent superposition of waves?

Let $\psi = A(t)\cos(\theta_1(t))$ and $\phi = B(t)\cos(\theta_2(t))$ two independent waves which phases and amplitudes depend on the time. Then it follows that the intensity of the superposition of ...
1
vote
1answer
67 views

Can a real harmonic function on the unit disk satisfy $f(0)=1$ while the area of $\{z:f(z)>0\}$ is zero?

Does there exist a harmonic function defined in the unit disk such that (1) $f(0)=1$ (2) area of $\{z\in\mathbb{D}:\,f(z)>0\}$ is equal to zero? I tried to use certain representations of ...
0
votes
0answers
8 views

Dirichlet with $C^1$ boundary data

Do you know an explicit example for an harmonic function $u$ in a regular domain $\Omega$ which satisfies the following two conditions? $u$ belongs to $C^0(\overline{\Omega})$ but not to ...
2
votes
1answer
19 views

An identity for the integral $ \int_{\partial B(0,1)}u(x_0+aw)u(x_0+cw)$ with a harmonic function $u$

This is Question 2.18 from Gilbarg and Trudinger, chapter 2. We are given that $\Omega$ is open bounded smooth boundary. Now fix $x_0\in \Omega$ and a constant $c>0$ such that ...
1
vote
1answer
20 views

Positivity of Poisson kernel

Let $K(x,y)$ be the Poisson kernel for the Dirichlet problem of Laplace equation on a bounded domain $D$ with smooth boundary, i.e. for a harmonic function $u$ on $\bar D$ with $u|_{\partial D}=g$, we ...
2
votes
1answer
40 views

Charge distribution on an arbitrarily shaped conductor

From physics we know that given a charged conductor put in vacuum (no external electric fields), the charge distribution on its surface is approximately proportional to the curvature of the surface on ...
0
votes
0answers
21 views

Physical interpretation of the integral formula for the solution of Laplace equation with Dirichlet/Neumann boundary condition

Suppose we have a bounded domain $D$ with smooth boundary, with $G(x,y)$ being the Green's function for the Poisson equation on $D$, i.e. $G(x,\cdot)=0$ on $\partial D$ and $\Delta_y ...
0
votes
1answer
29 views

Dirichlet problem on rectangle with nonhomogeneous boundary conditions

I want to solve the following problem in Dirichlet Problem: Let $D = \{(x, y) \in \mathbb R^2 : 0 < x < a; 0 < y < b\}$ be a rectangle with boundary $\partial D$. I want to solve ...
5
votes
1answer
166 views

Mean value property implies harmonicity

It is fairly easy to show that harmonic functions satisfy the mean value property, but it seems harder to show the converse. I've seen the following theorem without proof: If $u \in C(\Omega)$ ...
0
votes
1answer
70 views

You can't solve Laplace's equation with boundary conditions on isolated points. But why?

Consider a bounded region $\Omega\subset\mathbb R^n$ with a finite number of "holes" $X=\{x_1,\ldots,x_k\}$ that are isolated points in its interior. I'm pretty sure that in more than one dimension, ...
1
vote
1answer
39 views

Analytic function on the whole plane, positive imaginary part, what can it be?

Part (a): The function f is analytic in the whole plane with positive imaginary part. What can it be? Part (b): What if all you know is that the imaginary part of f tends to 0 at infinity? what we ...
3
votes
3answers
393 views

A positive harmonic function on the punctured plane is constant

Let $f(z)$ be a positive harmonic function on $\mathbb{C}\backslash \{0\}$. Prove that $f(z)$ is constant. I have no idea to prove this statement.
2
votes
0answers
32 views

Harmonic function with boundary value 0 except one point

Let $\Delta=\{z\in\mathbb{C}: |z|<1\}$ Assume $u\in C(\overline{\Delta}\setminus \{1\})$ such that it is harmonic in $\Delta$ and $u(\xi)=0$ for $\xi\in S^1\setminus \{1\}$. (a) Find an example ...
0
votes
1answer
146 views

Is my proof correct? (Invariance of subharmonicity under a conformal map)

I want to prove the following (exercise from Ahlfors' text): Prove that a subharmonic function remains subharmonic if the independent variable is subjected to a conformal mapping. Here is my ...
2
votes
1answer
490 views

Laplace heat equation

An infinite straight metal pipe has annular cross-section $a \leq r \leq b$. The temperature of the inner surface of the pipe is equal to $\cos(\theta)$, and the outer surface is thermally insulted. ...
0
votes
0answers
27 views

Prove that harmonic numbers satisfy the equality. [duplicate]

The $k$th harmonic number is defined to be $H_k = 1 + 1/2+ 1/3 + · · · + 1/k .$ Prove that harmonic numbers satisfy the equality $H_1 + H_2 + · · · + H_n = (n + 1)H_n − n$ for all $n \in\Bbb N.$
1
vote
2answers
73 views

Spherical harmonic expansion of a sphere

Seeing as one can expand any function on the sphere in terms of the spherical harmonics, I was thinking it should be possible to express the function for a sphere itself in terms of them. I have ...
4
votes
1answer
71 views

Isolated singularity of harmonic function

I am working with a book by Axler, Bourdon and Ramey and find the following problem: Suppose $u$ is a harmonic function on $B \setminus \{0 \}$ such that $$ |x|^{n-2} u(x) \to 0, \qquad x \to 0 $$ ...
0
votes
0answers
23 views

Pattern emerging from expansion of sphere in spherical harmonics

Seeing as one can express any function on the sphere in terms of the spherical harmonics, I am interested in what the sphere itself looks like when expanded. To get a function for the sphere, I use ...
0
votes
0answers
22 views

Harmonic function with Neumann BC

I have a problem where I denote by $D = B(0, 2)$ the disk in the plane with radius 2 centered at the origin. I have to find a harmonic function $u(r, \theta)$ in $D$ which satisfies the additional ...
0
votes
1answer
19 views

Harmonicity of the expectation of a stopped Brownian Motion

Let $\mathbb{E}_x$ be the expectation associated with a probability measure such that $B_{t\geq0}$ is a Brownian motion started in x. I want to show that for $D\subset\mathbb{R}^2$ bounded, $y\in D, ...
3
votes
0answers
36 views

Existence of a harmonic function?

I really have no clue about this problem. Should I represent $u(x)$ using $\varphi(x')$? But how to represent that? Thanks for any help!
1
vote
1answer
24 views

Is converse of Lewy theorem true?

In complex analysis, there is a result named Lewy's theorem, which states that: If $u=(u_1,u_2):\subseteq \mathbb{R^2}\to \mathbb{R}^2$ is one-one and harmonic in a neighborhood $U$ of origin ...
3
votes
2answers
50 views

There is no holomorphic function in $\Omega=\{0<r<\lvert z\rvert <R\}$ with real part $u(x,y)=\frac{1}{2}\log(x^2+y^2)$

Consider $u(x,y)=\dfrac{\text{log}(x^2+y^2)}{2}$ on $\Omega=\{0<r<|z|<R\}.$ Show there is no holomorphic function on $\Omega$ whose real part is $u.$ My attempt: I understand that $u$ ...
1
vote
0answers
18 views

Poisson Integral, when $U$ is discontinuous

So I am working on the following problem. Let $U$ be a piecewise continuous function and bounded for all real numbers. Then define the Poisson Integral for the UHP to be (It can be deduce from the one ...
0
votes
1answer
33 views

Dirichlet Problem

I have to solve the following Dirichlet Problem $$\Delta u=0\quad\text{in}\,\,\, D,$$ $$u(\mathrm{e}^{it})=\frac{1}{2}(\mathrm{e}^{it}+\mathrm{e}^{-it}),$$ for $$u \in C^2(D)\cap C(\overline{D}).$$ ...
8
votes
1answer
137 views

Area growth of harmonic functions

Can one construct a harmonic function $f$ defined in unit disk with condition $f(0)\geq1$ such that area of $\{z\in\mathbb{D}: f(z)>0\}$ is small enough?
1
vote
1answer
44 views

My attempt to prove an inequality get stuck——————where do I go wrong?

Hi, there. Bellow is my attempt. I don't know if I have gone in the wrong way and I am stuck. My attempt: Using Green's representation formula, $u(y)=\int_{\partial \Omega}u \frac{\partial ...
1
vote
3answers
37 views

Prove that $\phi(x,y)=e^{u(x,y)}cos(v(x,y))$ is harmonic

Suppose that $u,v$ are harmonic functions on doman $D$, and they are harmonic conjugate. Prove that function $\phi(x,y)=e^{u(x,y)}cos(v(x,y))$ is harmonic on $D$. What I've done was to take the ...
5
votes
1answer
117 views

Can the measure of zeroes of a harmonic function be positive?

Let $u$ be a non-constant harmonic function of two variables defined, say, in the unit disk (or on the half plane for example). It is known that $u$ can vanish on some lines, as it discussed in here. ...
0
votes
1answer
31 views

Strange thing about Weak Maximum\Minimum Principle?

I feel confused about this problem. I think it is obvious using Weak Maximum\Minimum principles. Since for harmonic functions. If $\Omega$ is bounded and $u\in C^2(\Omega) \cap C^0(\overline ...
0
votes
0answers
17 views

Why is the Laplace/Helmholtz equation only separable in a finite number of coordinate systems?

On MathWorld one finds that the Helmholtz equation $$(\nabla^2+k^2)\psi=0$$ is only separable in 11 coordinate systems. Similar statements can be found about the Laplace equation and maybe other ...
1
vote
1answer
25 views

Harmonic conjugate extend to boundary [duplicate]

Suppose u is a harmonic function in disc $|z|<1$, and u can be extended continuously to boundary, what about its harmonic conjugate v? Can it also be extended continuous to boundary? I know v can ...
1
vote
1answer
67 views

Uniqueness in boundary value problem for the biharmonic functions

My attempt: I tried to use the Green's representation formula twice. The Green's reprensentation formula:$u(y)=\int_{\partial \Omega}(u(x)\frac{\partial G(x-y)}{\partial v}-G(x-y)\frac {\partial ...
2
votes
1answer
32 views

Help with biharmonic functions

I tried to use using Green's formula to prove $\triangle u=0$. But it turned out to be wrong because $u=0$ doesn't imply $\triangle u=0$ on the boundary. I am a novice in graduate pde, so I didn't ...
0
votes
1answer
39 views

Local barrier implies barrier?

there. This is part of the textbook of Gibarg's PDE: My question is that how to verify the part in red? How to know $\overline w$ is continous in $\overline \Omega$? Thanks so much! Your help ...