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

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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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 ...
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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 ...
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2answers
346 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} \ \ ...
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1answer
261 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 ...
8
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1answer
70 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: $$ ...
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1answer
283 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 ...
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3answers
256 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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2answers
702 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.
6
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1answer
172 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 ...
6
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1answer
232 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
74 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 ...
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4answers
160 views

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
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3answers
1k views

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 $g$ ...
5
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1answer
198 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, ...
5
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2answers
68 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 ...
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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
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2answers
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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 ...
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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 ...
5
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1answer
279 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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3answers
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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
133 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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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 ...
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4answers
331 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 ...
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
214 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
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1answer
366 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 ...
4
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1answer
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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
449 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
48 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
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1answer
102 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 ...
4
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1answer
39 views

Showing that a given PDE is a solution to harmonic motion via transform

OK, here's an problem that should, by all accounts, be pretty simple, but I want to make sure I am approaching this correctly. Given: $$\frac{\partial ^2u}{\partial t^2}=c^2\frac{\partial ...
4
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2answers
174 views

Fourier transform of $|x|^{-t}$

In $\mathbb{R}^d$, let $f(x)=|x|^{-t}$, its Fourier tranform $F(f)(ξ):=(2\pi)^{-\frac{d}{2}}∫_{\mathbb{R}^d} e^{ix\cdot ξ}f(x)dx$, is there any fast way to see that this integral converge at $ξ \neq0$ ...
4
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1answer
182 views

Dirichlet Problem: Uniqueness of solution

Let $u$ be the solution to a Dirichlet Problem on a bounded open domain $D \subset \Bbb R^n$. Is the uniqueness of $u$ guaranteed by the maximum principle or by the smoothness of the boundary of $D$? ...
4
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2answers
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limit of harmonic functions Gilbarg and Trudinger page 27

Here is a passage from Gilbarg and Trudinger page 27 Let $\Sigma$ be a bounded domain in $\mathbb{R}^n$ ($n\geq 3$) with smooth boundary $\partial\Omega$, and let $u$ be the harmonic function (often ...
4
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1answer
199 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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2answers
142 views

Problem of Harmonic function.

If H is a harmonic function on an unit disk; And $H=0$ on $R_1\cup R_2$, here $R_1, R_2$ are radius of $D(0,1)$. The angle between $R_1$ and $ R_2$ is $r\pi$; here $r\in (0,1]$. If $r$ is an ...
4
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1answer
117 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 $u \in C(\Omega)$ ...
4
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1answer
97 views

Diffeomorphisms preserving harmonic functions

I'm looking for smooth maps $ \Bbb R^n \to \Bbb R^n $ with the property that, whenever $ h $ is a harmonic function ($\Delta h=0$), $ h\circ f $ is also harmonic. Is there a nice characterization of ...
4
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1answer
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Proof of weak maximum principle.

I read a proof in my book on pde which I find a bit strange. Let $\Omega$ be some bounded domain. For $f\in\mathscr C^2(\Omega)\cap\mathscr C^0(\Omega)$ satisfying $\Delta f\geq0$ it holds that ...
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1answer
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Harmonic functions in $\mathbb{R}^d$

I want to establish the equivalence of the 3 standard definitions, and that harmonic functions are $C^\infty$. The 3 definitions are: Mean value property and continuous. $C^2$ and $0$ Laplacian. ...
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1answer
385 views

Characterization of positive harmonic functions on unit disc with $0$ radial limits

Suppose $u$ is a positive harmonic function in $U$, and $u(re^{i\theta}) \to 0$ as $r \to 1$, for every $e^{i\theta} \ne 1$. Prove that there is a constant $c$ such that $$u(re^{i\theta}) = ...
4
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1answer
222 views

Simply connected domain and harmonic function

Let $\Omega$ be a simply connected domain that is properly contained in $\mathbb C$, and $u(x,y)$ is harmonic on the unit disk $\mathbb D $, then there is a funtion $f(z)$, that is one-one and ...
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0answers
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is Poisson's kernel always integrable?

Let $E$ be a smooth domain. The Green function $G(x,y)=F(x,y)-\Phi(x,y)$ where $F$ is the fundamental solution to the Laplace equation and for fixed $x\in E$, $\Phi(x,\cdot)$ is a harmonic function in ...
4
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1answer
79 views

Taylor expansions of harmonic functions.

Let $D \subset \mathbb{R}^2$ be the unit open disc. Note that any harmonic function on $D$ is real analytic. How can one prove that there exits a constant $C>0$ such that the Taylor expansion of ...
3
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3answers
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Laplace's Equation in Spherical Coordinates

The general solution of the Laplace equation in spherical coordinates is (independant of $\phi$): $$V(r,\theta ) = \sum ^{\infty} _{l=0} \left( A_l r^l + \frac{B_l}{r^{l+1}} \right) P_l (\cos \theta ...
3
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2answers
349 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.
3
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1answer
124 views

Visualization of subharmonic functions

I have always visualized subharmonic functions as Ahlfors' Complex Analysis thaught me to do: in one dimension lines are harmonic functions and "convex" functions are subharmonic. I actually just ...
3
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1answer
160 views

Finite order function in the complex analysis.

Assume that an entire function $f$ be finite order with finitely many zeros. Please show that either $f(z)$ is a polynomial or $f(z) + z$ has infinitely many zeros. Thank you. And I know the ...