For questions regarding harmonic functions.

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Trouble with an application of Green's representation formula

The teacher solved an exercise in class which required you to prove that, if $\Omega$ is a bounded domain in $\mathbb R^n$ and $G$ its Green function, then $G$ is symmetric, i.e. $G(x,y)=G(y,x)$ for ...
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0answers
23 views

Spherical harmonics: how's Laplace's equation related to spheres?

Many spherical harmonics derivations start from finding a solution to Laplace's equation and the results are in fact what are called spherical harmonics. However, how's Laplace's equation really ...
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1answer
23 views

What is wrong about this proof for the mean-value theorem for harmonic functions?

Let $\Omega\subset\mathbb{R}^n$ be an open connected domain, and let $u\in C^2(\Omega)$ be a harmonic function on $\Omega$. Then for every ball $B_R(x)=\{y\in\Omega:|x-y|<R\}$ in $\Omega$ we have $...
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1answer
21 views

Potential theory, potentials and harmonic functions

In the development of potential theory we mostly study harmonic functions. However I found some paper, which present potential theory as the study of potentials. Are potentials harmonic functions?
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29 views

Find solution of Laplace equation

Hey I need help with these example: Solve boundary problem on $\mathbb{R}^{+} \times \mathbb{R}^{+}$ \begin{equation*} \left\{ \begin{array}{l} \Delta u = 0 \\ u(0,.)=0 \\ u(.,0)= f\end{array}\right....
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1answer
19 views

square of polynomial still harmonic? [on hold]

Let $P(z)=\sum_{i=0}^n a_i z^i$ be a polynomials on $\mathbb{C}[z]$ such that $a_i$ are real numbers. $|P(z)|^2$ is a harmonic function ?
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1answer
1k views

Proof of strong maximum principle for harmonic functions

Let $u\in\mathscr C^2(U)\cap\mathscr C(\bar U)$ be harmonic in the non-empty open and connected set $U\subset\mathbb R^n$. If there exists a Point $x_0\in U$, so that $u$ has a local Maximum at $x_0$, ...
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1answer
38 views

Prove that $u(x,y) = \ln(x^2 + y^2)^{\frac{1}{2}}$ is harmonic on $\mathbb{C}\setminus \{0\}$ [closed]

Prove that $u(x,y) = \ln(x^2 + y^2)^{\frac{1}{2}}$ is harmonic on $\mathbb{C}\ {0}$, then find a harmonic conjugate $v(x,y)$ of $u(x,y)$ so that $f(z) = u(x,y) + iv(x,y)$ is analytic on $\mathbb{C}\...
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3answers
48 views

$u$ and $u^2$ are harmonic.

Let $D$ be the unit disk centered at $0$ in the complex plane, and let $u$ be a real harmonic function on D. Find all $u$ such that $u(0)=0$ and $u^2$ is also harmonic on $D$.
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Green's function for the Laplace or Helmholtz equations for an open rectangle in 2d or cylinder 3d?

I have only ever worked with free space Green's functions, or Green's functions for for the upper half space in 2d. So is it possible to determine a Green's function for the Helmholtz equation or ...
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0answers
12 views

Removing Singularity of real bounded harmonic in punctured disk. [duplicate]

If $u(z)$ is real harmonic and bounded in the punctured disk $0<|z-z_0|<R.$ Show that $\lim_{z\to z_0} u(z)$ exists. I already know Complex analytic function $f$ which has singularity $z_0$ ...
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0answers
14 views

Finite dimensional irreducible representations of Sp(2).

I am looking for an explicit description of the finite dimensional irreducible representations of the classical Lie group $\text{Sp}(2) = \{A\in M_2(\mathbb{H})\,|\,A\overline{A}^T = I\}$. I can ...
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0answers
15 views

An example of a bounded domain $\Omega\subset \left\{ 0<\Re s< 1\right\} $ for which $\Re \zeta(s)$ is non-negative

Denoting the complex variable $s=\sigma+it$ (and we know that $\mathbb{C}$ and $\mathbb{R}^2$ are isomorphic, thus $s\equiv(\sigma,t)\in\mathbb{R}^2$) one has for $0<\Re s=\sigma<1$ that $$\zeta(...
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0answers
33 views

Mean Value Property for harmonic functions

I would appreciate any insights on this matter; Let us consider the mean value property for harmonic functions in three dimensional euclidean space. This suggests that the value of a harmonic ...
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2answers
69 views

Construct a harmonic function that appears to be discontinuous on the unit circle.

Construct a harmonic function $u$ in $D(0,1)$ that satisfies $$ lim_{r \to 1^-}u(re^{i\theta}) = \begin{cases} 1 & 0 < \theta < \pi \\ 0 & \pi < \theta < 2\pi \...
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1answer
28 views

Does there exist a kernel concept for Taylor expansions?

In the sense of kernel which one can use when weighing coefficients together in Discrete Fourier Transform (Dirichlet and Fejér kernels maybe most well known). Here are some examples of such kernels. ...
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1answer
45 views

Decomposition of Harmonic function into sum of holomorphic and anti-holomorphic function

How do you prove that a harmonic planar mapping $f(x,y) = u(x,y) + i v(x,y)$ for real $u,v$ can be written as $f(x,y) = \phi(x,y) + \overline{\psi}(x,y)$ where $\phi$ is a holomorphic function, and $\...
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1answer
58 views

Prove: For a harmonic function $u$, $|z|^{1-\epsilon}|u(x,y,z)| \leq (x^2+y^2)^{0.5-\epsilon}$ implies $u = 0$ for $0 < \epsilon < 0.5$

Prove: For a harmonic function $u: \mathbb{R}^3 \rightarrow \mathbb{R}$, if for all $(x,y,z) \in \mathbb{R}^3$ $|z|^{1-\epsilon}|u(x,y,z)| \leq (x^2+y^2)^{0.5-\epsilon}$ then $u = 0$ for $0 < \...
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0answers
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Harmonic Function - Multivariable calculus

One more exercise I stepped at while strolling through papers and journals for my preparation on the semester exams for multivariable calculus. Let $D=\{(x,y): x^2 + y^2 \leq 1\}$ A function $f:D \to ...
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2answers
160 views

How to prove Liouville's theorem for subharmonic functions

I noticed this post and this paper, which gives a version of Liouville's theorem for subharmonic functions and the reference of its proof, but I think there must be an easier proof for the following ...
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1answer
77 views

Is $f(x)$ constant under these conditions?

Statement Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be an function that is concave up and increasing. If $\displaystyle \lim_{x\to \infty}\frac{f(x)}{x}=0$, then $f$ is constant. It'll be easy if ...
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2answers
45 views

Show : A holomorphic function is harmonic if $\frac{\partial f}{\partial \overline{z}}=0$

Let's consider a "new" basis of the partial differential operators (of order 1) on $\mathbb{R^2}\approx\mathbb{C}$ defined by : $\frac{\partial}{\partial z}:= \frac{1}{2}(\frac{\partial}{\partial x}-...
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0answers
31 views

Are harmonic functions ($\Delta u=0$) with compact support unique?

Does anyone how to solve the problem $$ \Delta u=0$$ with $u \in R^n$ and $u(x) \to 0$ as $|x| \to \infty$? Is $u=0$ the only solution? Many thanks.
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2answers
94 views

Find a solution that satisfies Laplace's equation in polar coordinates

How may I find a solution that solves Laplace's equation in polar coordinates, subject to the boundary conditions? In particular, I need to find one solution that satisfies $$\Delta u = 0,$$ subject ...
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0answers
71 views

Is there a mistake in this proof in Rudin's RCA?

Rudin's Real and Complex Analysis, 3rd edition, page 236 : How did Rudin get to $$ \frac{R-r}{R+r}u_n(a) \leq u_n(a+re^{i\theta}) \leq \frac{R+r}{R-r}u_n(a).$$ It seems to me that he used the mean ...
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1answer
67 views

Proof that $Y=A\cos(px)+B\sin(px)$ is only periodic if $p=n$.

I am asked to show that a function $y=A\cos(px)+B\sin(px)$ can only be periodic if $p$ is an integer $n$, where $A$, $B$ are arbitrary constants. In other words $y(x)=y(x+2 \pi)$ I begin by solving ...
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1answer
39 views

Harmonic forms and functions on compact manifolds

I hope my question is not stupid, I'm studying chapter 5.1 of Claire Voisin's book "Hodge theory and complex geometry". Let $X$ be a compact manifold and $A^k(X)$ be the space of $C^\infty$ forms on $...
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2answers
163 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 ...
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1answer
35 views

Show $u$ satisfy poisson equation

Let $f\in \mathcal S(\mathbb R^n),n\geq 3$. How can I show that $$u(x)=C_n\int_{\mathbb R^n}|x-y|^{2-n}f(y)dy$$ where $C_n$ is a suitable constant solve Poisson equation ? i.e. $$\Delta u=f.$$ The ...
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2answers
67 views

Methods for finding harmonic conjugate function

What are the methods for finding harmonic conjugate function? There is the cauchy - riemman equations but are there any other methods? Thank you very much
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1answer
25 views

A radial harmonic function on $\mathbb{R}^N \setminus \{0\}$ is of the form $\frac{b}{|x|^{N-2}} + c$

Prove: A radial harmonic function $f$ on $\mathbb{R}^N \setminus \{0\}$ is of the form $\frac{b}{|x|^{N-2}} + c$ for $b,c \in \mathbb{R}$. My try: Label $g_i = (0,..,0,x_i,0,..,0)$. From the maximum ...
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1answer
23 views

Is it possible to analytically solve Laplace's equation between two rectangles?

I need to solve the heat equation without sources (Laplace’s equation) on the green domain which is bounded by two rectangles shown below: Is it possible to do that analytically? So far I haven’t ...
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1answer
46 views

How to find the Green's function

Consider a domain $$D= \{(x,y) : x>0, y>0 \}$$ Let $\textbf x = (x,y)$ and $\xi =(\xi_x , \xi_y)$. Find the Green's function, $G(\textbf x , \xi)$ such that $$\nabla ^2 G=\delta (\textbf x - \xi)...
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1answer
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$f$ is holomorphic in Ω such that $|f|^2$ is harmonic; we need to show that $f$ is constant.

$f$ is holomorphic in Ω such that $|f|^2$ is harmonic; we need to show that $f$ is constant. solution of the question In the solution attached, I don't really understand the transition between $∆|f(z)...
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0answers
27 views

Non-constant harmonic function satisfying given property

Let $u(x,y)$ be a non-constant harmonic function in region $\mathbb{D}_{\mathbb{R}}=: D$ and $$A:=\{(x,y)\in D : u_x = u_y = 0\} $$ what can one say about the set $A$? Since $u$ is harmonic, there ...
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1answer
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A question related to Laplace equation on pde.

Let $B_R(p)$ the open ball of radius R centered at p in $R^n$ and consider the following problem $∆u = 0 $ in $ R^n\setminus B¯_R(p)$ ,$u = c$ on $∂B_R(p)$, where c is a given constant. Find a non-...
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2answers
36 views

What is a boundary condition for a PDE in a rectangular domain?

In the method of separation of variables, we need homogeneous BCs. For the elliptic pde with inhomogeneous BCs: $u_{xx}+u_{yy}=0$, with $0<x<a$ and $0<y<b$. With $u(x=0,y)=0$ and $u(x=a,...
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1answer
19 views

Geometry of level sets of an harmonic function

Suppose you have an harmonic function on an exterior domain of $\mathbb{R}^n$, i.e., a function $u \colon \mathbb{R}^n \setminus \bar\Omega \to \mathbb{R}$, where $\Omega$ is a smooth and bounded open ...
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Can the rank of harmonic map decrease far from the boundary?

This question is in some sense a continuation of this question, though it asks something weaker. Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we are ...
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Prove that a family of harmonic functions is a normal family

Suppose $\Omega$ is an open, bounded, connected set. Let $f$ be a continuous function on $\overline\Omega$ and $\mathcal{F}$ be the family of harmonic functions on $\Omega$ that belong to $C(\overline\...
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0answers
16 views

The harmonic functions are smooth

Let $u \in C^2(\Omega)$ a harmonic function, then $u \in C^{\infty}(\Omega)$. Here $\Omega$ is a domain of $\mathbb{R}^{n}$. Step 1 for proof: $u \equiv u^{\epsilon}$ in $\Omega_{\epsilon}$, for $\...
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1answer
130 views

About a harmonic function in the upper half plane [duplicate]

I'm struggling with the following question: Suppose that $C$ is a positive constant, $u$ is harmonic in the upper half plane $\mathrm{Im}z>0$, and that $0 \le u(z) \le C\mathrm{Im}z$ for $\mathrm{...
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0answers
43 views

Optimizing value of discrete harmonic function at a given point

Let $n>0$, and let $S_n$ denote the discrete square $S_n=[|-n,n|]^2$ (so $S_n$ has $(2n+1)^2$ elements). Let $K_n$ denote the set of four corner points $\lbrace (\pm n,\pm n)\rbrace$, and $C_n=S_n\...
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1answer
318 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 ...
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1answer
85 views

Existence of solutions to Laplace's equation for almost everywhere smooth boundary conditions

Let $\Omega$ be a compact region in the plane. Are there any existence results for the Dirichlet boundary value problem $$\begin{cases}\Delta f(q) = 0, & q\in \Omega\\ \lim_{p\to q} f(p) = g(q), &...
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0answers
61 views

Tight bounds for harmonic measure

I recently came across a question concerning harmonic measure here, and was wondering if there is a good reference summarizing different methods of estimating harmonic measure? Specifically, I would ...
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1answer
79 views

a special extension of a two variable function

We consider the function $f(x,y)=x^2+y^2$ in $\omega = (0,1)^2.$ I am wondering about the existence of a $C^2-$extension $F$ of $f$ in $\Omega = (0,2)^2$ such that $F$ is harmonic in $\Omega-\overline{...
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0answers
45 views

Equivalent definition of harmonic functions

Let $\Omega \subset \mathbb{R}^n$ be a domain. And let $u \in \mathcal{C}(\Omega)$. Prove that: The function u is harmonic in $\Omega$, i.e. $u \in \mathcal{C}^2(\Omega)$ and $\Delta u = 0$ on $\...
1
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1answer
27 views

Mean Value Property - Nonnegative Harmonic function

I want to prove that the mean value property $$u(\textbf{x}_0) = \frac{1}{\pi r^2} \int \int _{\left \{ \left | x_0-x \right < r| \right \}} u(\textbf{x})d\textbf{x}$$ for non-negative harmonic ...