For questions regarding harmonic functions.

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Why are harmonic functions called harmonic functions?

Are they related to harmonic series in any way? Or something else? Wikipedia didn't help.
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1answer
138 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 ...
11
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0answers
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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 $u$ ...
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2answers
396 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 ...
9
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1answer
431 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 ...
9
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2answers
429 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 ...
8
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2answers
356 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} \ \ ...
8
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1answer
146 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?
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1answer
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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: $$ ...
8
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1answer
456 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 ...
7
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1answer
254 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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1answer
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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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1answer
564 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
345 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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3answers
277 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: $$ ...
6
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3answers
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Composition of a harmonic function.

I noticed this question while reading several pdfs of lecture notes, and I'm having trouble figuring it out. Can anyone help? If $f$ is a harmonic function in a domain $D \subset \mathbb{C}$, and $g$ ...
6
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2answers
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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.
6
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1answer
277 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, ...
6
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2answers
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Showing that $P_r(x)=\frac{1-r^2}{1-2r\cos x+r^2}\rightarrow 0$ uniformly on $[-\pi,-\delta]\cup[\delta,\pi]$ as $r\uparrow 1$

Let $0<r<1$ and consider the series $$s = \sum_{n=-\infty}^\infty r^{|n|}e^{inx}.$$ I have shown that the series converges uniformely to $$P_r(x)=\frac{1-r^2}{1-2r\cos x+r^2}$$ on all of ...
6
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1answer
347 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 ...
6
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1answer
98 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 ...
6
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1answer
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oblique derivative smoothness of harmonic functions

Let $Q$ be a domain in the half-space $\mathbb R^n\cap\{x_n>0\}$ and part of its boundary is a domain $S$ on the hyperplane $x_n=0$. Let $u\in C(\bar Q)\cap C^2( Q)$ satisfy $\Delta u=0$ in $Q$ and ...
5
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4answers
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Compute $\left (1+\frac{1}{2}+\cdots+\frac{1}{n} \right )^2+\cdots+\left (\frac{1}{n-1}+\frac{1}{n} \right )^2+\left (\frac{1}{n} \right )^2$

Compute the value of the following expression $$\left (1+\frac{1}{2}+\cdots+\frac{1}{n} \right )^2+\left ( \frac{1}{2}+\cdots + \frac{1}{n}\right )^2+\cdots+\left (\frac{1}{n-1}+\frac{1}{n} \right ...
5
votes
4answers
423 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 ...
5
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2answers
94 views

Natural question about harmonic functions

Let $U$ an open, bounded and convex set in $R^n$. Let $(u_k)_{k \in N}$ a sequence of functions defined in $\overline{U}$. Suposse that each $u_k \in C(\overline{U}) \cap C^{2}(U)$ and harmonic in ...
5
votes
1answer
842 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 ...
5
votes
1answer
523 views

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 ...
5
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3answers
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Laplace's equation via Fourier transformation

I have a Laplace equation with some data along the $y$-axis: $$ \begin{cases} u_{xx} + u_{yy} &= 0 \\ u(0,y) &= f(y) \\ u_x(0,y) &= g(y). \end{cases} $$ There is no information of any ...
5
votes
1answer
132 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. ...
5
votes
2answers
89 views

Is this Harmonic Polynomial Identically Zero?

Let $h$ be a harmonic polynomial that is zero on the lines $Im(z) = 1$ and $Im(z) = -1$. I know that by the Schwarz Reflection Principle, $h$ must be zero on any line $Im(z) = k$ for $k$ odd, but ...
5
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1answer
228 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 ...
5
votes
1answer
240 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 $u \in C(\Omega)$ ...
5
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1answer
113 views

the real part of a holomorphic function on C \ {0, 1}

Let $h$ be a real valued harmonic function on the twice punctured plane $Ω = \text{C \ {0, 1}}$. Show that there exist unique real numbers $a_0$, $a_1$ such that $u(z) = h(z) − a_0 \log |z| − a_1 \log ...
5
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1answer
450 views

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 ...
5
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1answer
170 views

Topology of solution to a nonlinear eigenvalue problem

Consider the elliptic PDE: $$-\Delta u= f(x) u. $$ Assume that $f,u$ are defined in some reasonable bounded domain $\Omega \subset \mathbb{R}^n$ and impose the boundary condition $u=0$ on $\partial ...
5
votes
3answers
316 views

Laplace's equation on a square domain with a central point reservoire

Could someone please tell me the solution to this problem. I have $$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$$ on the square domain $-L<x<L, -L<y<L$ with ...
5
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1answer
171 views

Estimates on Derivatives

I have trouble in filling in the details of the proof on Estimates on derivates, from page 29 of PDE Evans, 2nd edition. Namely, I am lost at some steps. The book gives: Theorem 7 (Estimates on ...
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0answers
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Finding the harmonic conjugate

I have shown that the following function is harmonic and am attempting to find it's harmonic conjugate: $u=e^{-2xy}\sin(x^2-y^2)$ I know that to find the harmonic conjugate I need to use the ...
4
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1answer
239 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, ...
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2answers
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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 ...
4
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2answers
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Green's function

Why does the Green's function $G(r,r_0)$ of the Laplace's equation $\nabla^2 u=0$, the domain being the half plane, is equal $0$ on the boundary? How can I interpret the Laplace's equation physically? ...
4
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2answers
229 views

harmonic function question

Let $u$ and $v$ be real-valued harmonic functions on $U=\{z:|z|<1\}$. Let $A=\{z\in U:u(z)=v(z)\}$. Suppose $A$ contains a nonempty open set. Prove $A=U$. Here is what I have so far: Let ...
4
votes
1answer
197 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 ...
4
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1answer
272 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 ...
4
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3answers
887 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 ...
4
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1answer
74 views

Determining if a Continuous $u:\mathbb{C}\to \mathbb{R}$ Satisfying some Property is Harmonic

If $u : \mathbb{C} \to \mathbb{R}$ satisfies $$u(x + iy) =\frac{1}{4}[u(x + a + iy) + u(x − a + iy) + u(x + i(y + a)) + u(x + i(y − a))]\tag{$*$}$$ for all $a$ then determine whether $u$ is harmonic, ...
4
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1answer
239 views

Dirichlet problem in the disk: behavior of conjugate function, and the effect of discontinuities

Dirichlet's problem in the unit disk is to construct the harmonic function from the given continuous function on the boundary circle. It is solved by the convolution with the Poisson kernel, and we ...
4
votes
1answer
616 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 ...
4
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1answer
58 views

Boundary data of the modulus of a holomorphic function

Let $f$ be a non-vanishing holomorphic on the unit disk $D$. Suppose $|f|$ converges to a measure $\mu$ on $\partial D$ as $|z|\rightarrow 1$, in the sense that $$ \int_{\partial D} |f(r z)| \phi(z) ...
4
votes
1answer
150 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 ...