Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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19
votes
2answers
603 views

Is it Possible to Construct all Proofs in Complex Analysis using Brownian Motion?

(First, I am very aware of the fact that Brownian motion is actually probably more difficult to understand than at least basic complex analysis, so the pedagogical merits of such an approach would be ...
18
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2answers
2k views

Why are harmonic functions called harmonic functions?

Are they related to harmonic series in any way? Or something else? Wikipedia didn't help.
14
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4answers
8k views

Composition of a harmonic function with a holomorphic function is still harmonic

If $f$ is a harmonic function in a domain $D \subset \mathbb{C}$, and $g$ is a conformal mapping of a domain $D_0$ onto $D$, is $f \circ g$ harmonic in $D_0$? I noticed this question while reading ...
12
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1answer
2k views

Why is it important to study the eigenvalues of the Laplacian?

Why is it important to study the eigenvalues of the Laplacian acting on regions in $\mathbb R^n$? What information does this give us? What problems does this information help us solve? In particular, ...
12
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1answer
3k views

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 ...
11
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2answers
5k views

Logarithm of absolute value of a holomorphic function harmonic?

Let $f:U\rightarrow\mathbb{C}$ be holomorphic on some open domain $U\subset\hat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$ and $f(z)\not=0$ for $z\in U$. Is it true that $z\mapsto \log(|f(z)|)$ is ...
11
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1answer
189 views

Source of the “$\cosh$ trick” for Laplacian eigenfunctions or Helmholtz equation solutions?

Suppose a smooth function $f : \mathbb{R}^n \to \mathbb{R}$ satisfies the Helmholtz equation, the PDE $\Delta f + k^2 f = 0$. A while ago someone showed me a trick: Define a function $g:\mathbb{R}^{n+...
11
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0answers
787 views

Kato inequality [closed]

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 $u$ ...
10
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2answers
2k views

Show harmonic function is constant on $\mathbb{R}^n$

I'm trying to solve the following question (this is just for practice): If $u$ is harmonic within $\mathbb{R}^n$ with $\int_{\mathbb{R}^n}|Du|^2 dx \leq C$ for some $C > 0$, then show that u is ...
10
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2answers
673 views

Integration trick $\int^{2\pi}_{0} f(a+ r \cos \theta, b+r\sin \theta)d\theta=2\pi f(a,b)$

While looking for some nice integrals that are not taught in school, I found this theorem: Suppose $f$ is a bivariate harmonic function; $(a,b)$ is a point in the plane; and $r$ is a positive real ...
10
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1answer
915 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 self-study -...
10
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1answer
3k views

If $u$ is harmonic and bounded in $0 < |z| < \rho$, show that the origin is a removable singularity

This is a reworking of a previous question here which was marked as a duplicate. Some nice folks have referred me to solutions to similar problems. I still have a couple of questions, since one of the ...
10
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2answers
1k 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 $u$...
10
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2answers
29k views

Calculating a harmonic conjugate

Is the following reasoning correct? Determine a harmonic conjugate to the function \begin{equation} f(x,y)=2y^{3}-6x^{2}y+4x^{2}-7xy-4y^{2}+3x+4y-4 \end{equation} We first of all check if $f(x,y)$ ...
10
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1answer
5k views

What is the Fourier transform of spherical harmonics?

What is the definition (or some sources) of the Fourier transform of spherical harmonics?
10
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1answer
205 views

Is there more to complex analysis than a two-dimensional potential theory?

Wikipedia entry on Potential theory in two dimensions says the following From the fact that the group of conformal transforms is infinite-dimensional in two dimensions and finite-dimensional for ...
10
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1answer
1k views

Error on Wikipedia: Nelson's proof of Liouville's theorem works only for bounded modulus?

On Wikipedia, it is stated: If $f$ is a harmonic function defined on all of $\mathbb{R}^n$ which is bounded above or bounded below, then $f$ is constant...Edward Nelson gave a particularly ...
9
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2answers
7k 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.
9
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3answers
298 views

Heat Equation + Uniform Convergence in time -> Harmonic Limit

Assume we have $u \in C^3(\mathbb{R}^n \times (0,\infty))$ satisfying the heat equation $$ \Delta u(x,t) = \partial_t u(x,t)$$ and a function $u_0:\mathbb{R}^n \to \mathbb{R}$ with unknown regularity (...
9
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3answers
397 views

Is there a reason why Harmonic functions are defined on open sets?

Whenever I see a definition of a harmonic function, it's always defined as follows A function $f : U \to \Bbb{R}$ is called harmonic (where $U$ is an open subset of $\Bbb{R}^n$) iff it is twice ...
9
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1answer
1k views

Finding fundamental solution to the biharmonic operator $\Delta^2=\Delta(\Delta)$?

Show that when $n=2$, the function $u(x)=-\dfrac{1}{8\pi}\left\lvert x\right\rvert^2\log\left\lvert x\right\rvert$ is a fundamental solution to the biharmonic operator $\Delta^2=\Delta\left(\Delta\...
9
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1answer
301 views

Green's identity contradicts Helmholtz theorem

Let F be a vector field on a bounded domain $V \subset \mathbb{R}^3$, which is twice differentiable, let $S := \partial V$. According to the Helmholtz Theorem, F can be decomposed, such that: $$ \...
9
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1answer
832 views

Approximation of a harmonic function on the unit disc by harmonic polynomials.

Let $u$ be a real valued harmonic function on the open unit disc $D_1(0) \subseteq \mathbb{C}$. Show that there exists a sequence of real valued harmonic polynomials that converges uniformly on ...
9
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1answer
414 views

Solutions to Dirichlet problem on manifolds with boundary

I am looking for a reference for the following assertion: Let $M$ be a Riemannian manifold with boundary, and $f:\partial M \rightarrow \mathbb{R}$ be smooth. Then there exists a unique smooth ...
8
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4answers
164 views

Prove that inequality $1+\frac{1}{2\sqrt{2}}+…+\frac{1}{n\sqrt{n}}<2\sqrt{2}$

Let $n$ is a natural number. Prove that inequality $$1+\frac{1}{2\sqrt{2}}+\frac{1}{3\sqrt{3}}+...+\frac{1}{n\sqrt{n}}<2\sqrt{2}$$ My try: $$\frac{1}{n\sqrt{n}}=\frac{\sqrt{n}}{n^2}<\frac{\sqrt{...
8
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2answers
558 views

If a Laplacian eigenfunction is zero in an open set, is it identically zero?

This seems like it should be easy but I can't seem to figure it out. Suppose $f$ satisfies $\Delta f + \lambda f = 0$ in $\mathbb{R}^n$ (where $\lambda > 0$ is a constant), and in some open set $U$...
8
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1answer
576 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 ...
8
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2answers
383 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} \ \ \partial\Omega....
8
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1answer
484 views

(Rudin's) Definition of a harmonic function

In Chapter 11 of Rudin's RCA, a harmonic function is defined to be a complex continuous function $u$ on a plane open set such that the Laplacian of $u$, i.e. the sum of its pure second-order partial ...
8
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1answer
675 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 ...
7
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2answers
1k views

Prove that a non-constant harmonic function is an open map.

I'm trying to solve the following exercise of the book Functions of one complex variable, John B. Conway on page 255: 4. Prove that a harmonic function is an open map. (Hint: Use the fact that the ...
7
votes
1answer
2k 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 $u \in C(\Omega)$ ...
7
votes
1answer
266 views

Construction a harmonic function

Suppose $f:\mathbb{R}\to\mathbb{R}$ is a smooth, positive, bounded function on $\mathbb{R}$. Construct a real-valued continuous function $u$ on $\overline{\mathbb{H}}$ which are harmonic on $\mathbb{H}...
7
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1answer
2k views

Laplace's Equation with Neumann BC

Hi fellow math enthusiasts, I am currently working on some research to do with the electric field induced within the brain via magnetic stimulation. I am trying to solve the partial differential ...
7
votes
1answer
974 views

Complex and real forms of the Poisson integral formula

In my complex analysis book there is the expression $$\frac{1 - |z|^2}{|1 - \bar z e^{it}|^2}$$ and it says that when $z = re^{it}$, we can write the above expression as $$P_r(t) = \frac{1 - r^2}{1 - ...
7
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1answer
180 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?
7
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1answer
365 views

counterexample for ill posedness of the laplace equation

Consider the wave equation with initial data: $$u_{tt}(t,x) + u_{xx}(t,x) = 0$$ $$u(0,x) = u_0(x)$$ $$u_t(0,x) = u_1(x)$$ Hadamard showed that this problem is ill-posed: there exist large solutions ...
7
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1answer
65 views

Can we approximate harmonic functions with harmonic functions with non-vanishing differential?

Let $\mathbb{D}^2$ be the closed $2$-dimensional unit disk, and let $g:\mathbb{D}^2 \to \mathbb{R}$ be a non-constant harmonic function (in particular smooth up to the boundary). Does there exist a ...
7
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3answers
446 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: $$ Tf(...
7
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0answers
446 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 ...
6
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4answers
1k 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 ...
6
votes
2answers
478 views

Analytic continuation of harmonic series

Is there an accepted analytic continuation of $\sum_{n=1}^m \frac{1}{n}$? Even a continuation to positive reals would be of interested, though negative and complex arguments would also be interesting. ...
6
votes
3answers
293 views

Bounded Harmonic Functions on the Disk

Denote by $\mathbb{D}$ the open unit disk in $\mathbb{R}^2$. Is it possible to find a bounded harmonic function $u : \mathbb{D} \to \mathbb{R}$ that is not uniformly continuous? I tried using ...
6
votes
3answers
12k 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 ...
6
votes
1answer
682 views

Solving laplace's equation for an inviscid and incompresible fluid

Background I'm working on a 2D inviscid, incompressible fluid sim using vortex methods (that is, treating vortex as discrete particles), and I'm trying to (numerically) solve the no-through boundary ...
6
votes
1answer
67 views

Let $u \in C^3(\mathbb{R}^n)$ and harmonic such that $u(x) = o(|x|)$ when $|x| \rightarrow \infty$. Show that $\nabla u(0) = 0$ and u is constant

Let $u \in C^3(\mathbb{R}^n)$ and harmonic such that $u(x) = o(|x|)$ when $|x| \rightarrow \infty$. 1. Show that $\nabla u(0) = 0$ 2. Show that u is a constant function. To show that $\nabla u(0) = 0$...
6
votes
2answers
110 views

Are spherical harmonics harmonic?

According to Wikipedia, a harmonic function is one which satisfies: $$ \nabla^2 f = 0 $$ The spherical harmonics (also according to Wikipedia) satisfy the relation $$ \nabla^2 Y_l^m(\theta,\phi) = -\...
6
votes
1answer
213 views

Mean value type property

It is well known that every harmonic function satisfy the mean value property. I was wondering if there exists some similar property for functions $u \colon \Omega \subset \mathbb{R}^n \to \mathbb{R}$ ...
6
votes
1answer
506 views

Harmonic function in a unit disk with jump boundary data

I am reading Conway's book about complex analysis. One question in it bothered me a lot recently. If given a piecewise continuous function with jump on the boundary of unit disk and it is bounded, we ...
6
votes
1answer
752 views

Interior gradient bound

I would like some help with the following problem (Gilbarg/Trudinger, Ex. 2.13): Let $u$ be harmonic in $\Omega \subset \mathbb R^n$. Use the argument leading to (2.31) to prove the interior ...