For questions regarding harmonic functions.

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1answer
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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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0answers
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Trobule 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

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
18 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
37 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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0answers
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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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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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0answers
32 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 ...
1
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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
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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0answers
51 views

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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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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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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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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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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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
24 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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2answers
60 views
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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
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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1answer
73 views

$f$ 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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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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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
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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0answers
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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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0answers
22 views

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
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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
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 ...
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2answers
22 views

Fourier series: can a function be odd and have a dc component?

Long story short: fourier series is taken in two subjects (for now). One doc says that the dc component is 0 if the function is odd. The other says that odd and even has no effect on the dc ...
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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 $\...
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1answer
36 views

Uniform Convergence: Poisson Kernel

If we fix $θ_∗ > 0$, then $P(r, θ) → 0$ uniformly on the set $ \left\lbrace θ : |θ| ≥ θ_∗ \right\rbrace $ as $r → a^-$ $$P(r,\theta) = \frac{a^2-r^2}{a^2-2r\cos(\theta)+r^2}$$ $0\leq r <a,\...
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1answer
30 views

Plotting Spherical Coordinates

I'm trying to plot the Poisson Kernel, where a = 1, so the resulting equation would be $$P(r,\theta) = \frac{1-r^2}{1-2r\cos(\theta)+r^2}$$ $0\leq r <a = 1 ,\textrm{ } -\pi \leq \theta \leq \pi$ ...
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0answers
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Harmonicity on semisimple groups

Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in $U(\mathfrak{g})$ and let $f$ be a smooth (or analytic) real-valued ...
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1answer
21 views

Bounds on derivatives of harmonic functions on unit ball

Let $u$ be a harmonic function on the unit ball in $\mathbb{R}^n$. Show that $$\sup_{B_{1/2}} \lvert \nabla u \rvert \leq C(n) \sup_{\partial B_1} \lvert u \rvert$$ More generally, show that $$\...
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0answers
13 views

Mean value property for harmonic functions (geometric idea)

Someone knows a bibliographic reference geometrically explore the idea of ​​the mean property for harmonic functions in domains of $\mathbb{R}^{n}$ ?
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2answers
36 views

Newton potential for Neumann problem on unit disk

Problem: Show that $$G(x, y) = \frac{1}{2\pi} \left( \log \lvert x - y \rvert + \log \left\lvert \frac{x}{\lvert x \rvert} - \lvert x \rvert y \right\rvert \right)$$ is a Green's function for the ...
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0answers
16 views

Find the complex potential given the real part $\frac{6000}{\pi}\theta$

In my 'Complex Variable Theory' notes, we have an example where we use a linear fractional mapping to solve for Laplace's equation between semi-circular plates. The Question reads: Find the ...
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2answers
158 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
15 views

Are there identities which will make calculating $\Delta \frac{e^{ik|x|}}{|x|}$ more efficient and clearer?

Let $x = (x_1, x_2, x_3)$ be a point in $\mathbb{R}^3$. I am trying to calculate $$\Delta \frac{e^{ik|x|}}{|x|}$$ Doing this 'manually' be calculating second derivatives is really long and tedious ...
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2answers
27 views

Laplace transform

Hello how can I solve this problem $ty''-y'=t^2$, $y (0)=0 $ I though about several ways like $y''-y'/t =t$ but I did not know how to solve the integral of $( y'/t) $. Then I thought about ...