For questions regarding harmonic functions.

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4
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1answer
67 views
+50

$\frac{1}{2\pi}\int_0^{2\pi}\log|re^{it}-\zeta|dt$?

Prove that if $\zeta \in \mathbb{C}$ and $r>0$ then $\frac{1}{2\pi}\int_0^{2\pi}\log|re^{it}-\zeta|dt=\log |\zeta|$ if $r\leq |\zeta|$, and it is $\log r$ if $r> |\zeta|$. My Try: First I ...
0
votes
2answers
38 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 ...
0
votes
0answers
29 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.
0
votes
0answers
29 views

How to prove a function is not positive definite [on hold]

I have a lecture about matrix analysis. I have already know some strategies to prove that the function is positive definite. But I face difficulties when I try to see that the (bounded) function is ...
2
votes
2answers
91 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 ...
1
vote
0answers
69 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 ...
0
votes
1answer
65 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 ...
0
votes
0answers
29 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 < ...
1
vote
1answer
37 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 ...
14
votes
2answers
138 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 ...
2
votes
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 $x_0$, ...
0
votes
1answer
34 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 ...
0
votes
2answers
66 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
0
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1answer
22 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 ...
0
votes
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 ...
0
votes
1answer
42 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 - ...
-2
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2answers
59 views
3
votes
1answer
66 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 ...
0
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0answers
26 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 ...
0
votes
1answer
16 views

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 ...
0
votes
2answers
33 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 ...
2
votes
1answer
12 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 ...
1
vote
0answers
25 views

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 ...
0
votes
0answers
21 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 ...
0
votes
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 ...
1
vote
1answer
123 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 ...
5
votes
0answers
41 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 ...
4
votes
1answer
316 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 ...
0
votes
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), ...
2
votes
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 ...
1
vote
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 ...
3
votes
1answer
37 views

$g: (U \times U - D) \to \mathbb{R}$ is continuous, $D$ diagonal? [closed]

Do we have necessarily have that$$g: (U \times U - D) \to \mathbb{R},$$is continuous, where $D$ is the diagonal? Idea. Perhaps we want to apply the maximum-minimum principle to $G(z, z_0)$?
0
votes
0answers
43 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
vote
1answer
24 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 ...
0
votes
2answers
21 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 ...
1
vote
1answer
33 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 ...
0
votes
1answer
30 views

Plotting Spherical Coordinates

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

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 ...
0
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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 ...
1
vote
1answer
20 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 ...
0
votes
0answers
42 views

Is this a right calculation of Fourier transform?

I am trying to solve: Calculate Fourier transform (on $-\infty < x < \infty $) of: $$ f(x) = \begin{cases} 1, & -L<x<L \\ 0, & |x|\geq L \end{cases} $$ My ...
0
votes
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 $\mathbb{R}^{n}$ ?
0
votes
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 ...
0
votes
2answers
110 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 ...
1
vote
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 ...
-1
votes
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 ...
0
votes
0answers
17 views

Stuck while trying to show fundamental solution $\frac{1}{2\pi}\ln|x|$ satisfies $\Delta \phi(x) = \delta(x)$ in 2d?

I am trying to show that the fundamental solution to the Laplacian in 2D satisfies $$\Delta \phi(x) = \delta(x)$$ where $x = (x_1, x_2) \in \mathbb{R}^2$. So the fundamental solution in 2D is ...
2
votes
1answer
59 views

Separation of variables for PDE: dividing by zero?

This feels like a question that is both simple and duplicate but I can't find an answer or a previous version of the question. Suppose we are given some PDE, for example Laplace's Equation in polar ...
3
votes
1answer
101 views

Harmonic function with injective boundary conditions is an immersion?

This question is in some sense a continuation of this question, though more focused in its scope. Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we ...
2
votes
1answer
34 views

Properties of harmonic function on $\mathbb{R}^2$

Assume $f$ is harmonic on $\mathbb{R}^2$. I want to prove that if there exists a constant $M$ such that $f(x,y) \geq M$ for all $(x,y)\in \mathbb{R}^2$, then $f$ must be a constant fuction. I'm ...