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)

1
vote
1answer
16 views

Derivation of energy integral - harmonic functions

I am following the solution of the following problem on the topic of the energy integral of a surface. For a real-valued continuously differentiable function $u(x,y)$ on a closed domain $D$, the ...
0
votes
1answer
19 views

The tangent hyperplane to the graph of harmonic function

This is an interesting question I found online about Laplace equation. We define a function $u$ :$R^N\to R$'s graph to be the set $\{x,u(x):\,x\in R^N\}\subset R^{N+1}$. Then I want to prove that ...
2
votes
1answer
29 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 ...
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 ...
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 ...
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
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 ...
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 ...
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
20 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 ...
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
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 ...
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 ...
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 ...
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 ...
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
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, ...
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 $n \in\Bbb N.$
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 ...
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 ...
0
votes
1answer
27 views

Harmonic function with Neumann boundary condition in the disk

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 ...
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 $$ |x|^{n-2} u(x) \to 0, \qquad x \to 0 $$ ...
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 ...
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!
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 $$u \in C^2(D)\cap C(\overline{D}).$$ ...
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 ...
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?
0
votes
1answer
32 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 ...
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. ...
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

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 ...
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 ...
1
vote
0answers
44 views

Prove a function is subharmonic?__Gilbarg Exercise

My attempt: By a long time google search I found in a book about how to prove it in the case of harmonic function. Just construct another function $v$ which is harmonic in the $\Omega$, then prove ...
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 ...
1
vote
1answer
41 views

Find a harmonic function on two concentric balls?

My attempt: I thought about using Poisson Integral formula since the area is two concentric balls. Then I get something like the following: $u(x)=\frac{1}{nw_nR}\int_{\partial ...
1
vote
1answer
46 views

The solution of $\Delta u=u^3$ with zero boundary values is identically zero

My question: My attempt: I tried to use the Representation using Green's formula: Since $u=0$ on the boundary and $f(x)=x^3$, then the formula becomes: $$u(x)=\int_\Omega y^3G(x,y)dy \quad ...
0
votes
1answer
27 views

Harmonic functions that uniformly convergent?

Let $u_k$ be continuous on $\overline\Omega$, $u_k$ harmonic in $\Omega$. Suppose $u_k|\partial\Omega$ converge uniformly. Then $u_k$ converge uniformly in $\Omega$. The hint is using Maximum ...