For questions regarding harmonic functions.

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How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)?

We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and ...
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81 views

Function must be constant comparison

The following 2 problems are past exit exam problems for my major. I see that they're worded differently but are asking me to do the same thing. Not sure how they differ much I'd appreciate if anyone ...
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53 views

Biharmonic boundary condition

I try to solve $$\Delta^2u=f$$ on unit square. with $f=4sin(\pi x)sin(\pi y)$ Using $v=-\Delta u,$ leads to $$v+\Delta u=0,$$ $$-\Delta v=f.$$ By Dirichlet boundary condition on $u$. What is ...
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176 views

Geometric Interpretation of Laplace's Equation

Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be analytic. In the natural way, let $f = u + vi$ for $u,v : \mathbb{C} \rightarrow \mathbb{R}$. Let $z \in \mathbb{C}$. Suppose that $u$ and $v$ satisfy ...
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Harmonic function [duplicate]

Let $B(0;1)=\{x \in \mathbb{R}^N;|x|≤1\}$, the ball of $\mathbb{R}^N$ equipped with the euclidian scalar product $$x⋅y=x_1y_1+...+x_Ny_N,\ \ \ x=(x_1,...,x_N),\ \ \ y=(y_1,...,y_N)\ \ \ |x|=\sqrt{x ...
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1answer
118 views

Choose parameters to make a harmonic function

Let $B(0;1)=\{x\in \mathbb{R}^N ;|x|\leq 1\}$, the ball of $\mathbb{R}^N$ equipped with the euclidian scalar product $$x \cdot y=x_1y_1+...+x_Ny_N,\ \ \ x=(x_1,...,x_N),\ \ y=(y_1,...,y_N)\ \ \ ...
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114 views

Why don't elliptic PDE's have a time coordinate?

Usually second-order linear PDE's are classified as elliptic, parabolic, or hyperbolic (or ultrahyperbolic) depending on the eigenvalues of the coefficient matrix. The three cases correspond to the ...
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35 views

$\{u_{n}\}$ harmonic and converging uniformly to $u \Rightarrow $ $u$ harmonic

Let $A$ be an open set, $\{u_{n}\}$ a sequence of harmonic functions on $A$, converging to $u$ uniformly on compact subsets of $A$ . Then $u$ is harmonic on $A$. Any hint ?
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1answer
215 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, ...
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1answer
124 views

Finding the Boundary Conditions for a Laplace's Equation in Polar Coordinates

I have solved Laplace's equation in Polar Coordinates for the scalar electric potential in a circle of radius R and have the solution $$ \phi(r,\varphi) = \phi_{0} + ...
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62 views

implicit derivates incorperating laplace's equation

If $f(x,y)$ is a harmonic function show that the function $F(x,y)=f(x^2-y^2,2xy)$ is also harmonic. You have to use Laplace's formula to prove this, unless there is an easier way. I'm having trouble ...
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54 views

Harmonic functions constant on circumferences

I want to find all the harmonic functions in $\mathbb{R}^{2}-\{(0,0)\}$ which are constant on circumferences with center in $(0,0)$. $\mathbb{R}^{2}-\{(0,0)\}$ isn't simply connected so we can't ...
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1answer
64 views

polar Laplace equation solution:

Question: $ \frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{1}{r^2}\frac{\partial^2 u}{\partial {\theta}^2} = 0,\:\:\:\: 0\leq r \leq 3 \:\:\:\: ...
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1answer
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Bounded (from below) harmonic functions from $\mathbb R^2 \setminus \{0\}$

Let $M \in \mathbb R$ be a real number and $u\colon \mathbb R^2 \setminus \{0\} \to [M, +\infty)$ be an harmonic function. Then it is constant. Show that this is no longer true in higher ...
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1answer
33 views

Examples for 2-dimensional real valued harmonic functions

Given: $$f: \mathbb{R}^2 \to \mathbb{R},\space \Delta{f}=0, \space\frac{\partial^2{f}}{\partial{x}^2_i} \neq 0, i \in\{1,2 \}.$$ Are there examples of such functions?
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55 views

One problem about harmonic functions

Problem. Given open, bounded set $\Omega\subset\mathbb R^d$ with smooth boundary $\partial\Omega$ and given smooth function $\varphi$ on $\partial\Omega$. As known, problem $$ \begin{cases} ...
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168 views

$uv$ is harmonic if and only if $u+icv$ is analytic for some real c

Let $u$ and $v$ be non constant harmonic functions on a complex domain. Prove that $uv$ is harmonic if and only if $u+icv$ is analytic for some real $c$. I can prove the "if" part. I am having some ...
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1answer
152 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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214 views

Uniqueness of harmonic solution

On page 28, line 1 in PDE Evans, 2nd edition, the theorem and proof state Theorem. Let $g \in C(\partial U), f \in C(U)$. Then there exists at most one solution $u \in C^2(U) \cap C(\bar{U})$ of ...
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343 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 ...
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What is the difference between these two kernel definitions?

I am reading my graft and the document of David Haussler about Convolution Kernels on Discrete Structures, UCSC-CRL-99-10. My graft and the other document The terminology seems to differ. The ...
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4answers
165 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 ...
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166 views

Dirichlet Problem on the unit disk

Find a C-harmonic function in the unit disk with boundary values $x^3-xy$. I know the answer is $u(x,y)=\frac{(x^3-3xy^2)}{4} + \frac{3x}{4} - xy$ but don't know how to solve it Any hint or help is ...
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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 ...
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1answer
156 views

Does the Poisson kernel give a unique harmonic function with given boundary data?

In the answer to this question, a helpful Stack user said that the Poisson kernel does not necessarily give a unique harmonic function, given certain boundary data (in particular on the upper ...
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1answer
136 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 ...
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1answer
101 views

Upper bound for coefficients of a power series

I am doing the following problem. Suppose $f(z)=\sum_{n=0}^{\infty}a_nz^n$ is an analytic function on the unit disc $|z|<1$. Let $0<r<1$. Prove that $$|a_n|r^n\leq \max\{4A(r),0\}-2Ref(0),$$ ...
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171 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 ...
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1answer
79 views

Harmonic function reflection

I'm learning some harmonic function theory by reviewing some problems. I came across two: 1) Prove that a real harmonic function $u$ from $\mathbb{R}^n$ to $\mathbb{R}$ such that $u(x, 0) = 0$ for ...
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Find a function $f$ such that $f$ is harmonic on $D$ and $f|_{\partial D}$.

I understand its solutions in general. But my question is how to decide whether I sould take $Im z^4$ or $Re z^4$? I have two similar examples. And in one example, the real part is taken, but in ...
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1answer
182 views

Derive Poisson integral formula in a ball

Trying to derive by myself the Poisson integral formula in a unit ball. I should get $$\Delta u=0 \,\text{ in } B(0,1), \,\,\, u(x)=\varphi(x)\,\,\text{at } \partial B(0,1) \Longrightarrow \\$$$$u(x) ...
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1answer
348 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 ...
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472 views

Poisson Integral Formula

I'm looking at the following problem. Prove that if $h$ is harmonic on an open neighborhood of the disc $B(w,\rho)$, then for $0 \leq r < \rho, 0 \leq t < 2\pi$, $$h ...
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1answer
152 views

Laplace equation on unbounded set

I have gotten stuck with a problem for PDEs class for a few days. I did not figure out how to start a solution for it. Problem: Let $g \in C(\partial B(0, R))$, $n > 2$. Find a formula for a ...
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A question on Harmonic functions.

$f$ and $g$ are two given continuous functions on $\overline {D(0,1)}$) and both $f$, $g:\overline {D(0,1)}\to\Bbb R$ are harmonic on unit disc $D(0,1)$. Now, If $f\equiv g$ on some $C=\{e^{i\theta} ...
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246 views

Show that the Kelvin-transform is harmonic

First, I have to give you our definitions: Consider $\Omega:=\mathbb{R}^n\setminus\overline{B}_R(0)$ with $R>0$ and $n>1$. The function $\phi\colon\Omega\to G:=B_R(0)\setminus\left\{0\right\}$ ...
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2answers
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The conditions for Harmonics functions in complex analysis

Let $\sigma(u, v)$ and $\gamma(u, v)$ be harmonic functions on a region $D$ in $\Bbb C$. What are the conditions on $\sigma$ and $\gamma$ such that $\sigma \gamma$ is harmonic on $D$. And I want to ...
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386 views

Reflection principle for harmonic functions

$U^+ \colon= \left\{x\in \mathbb{R}^n\mid |x| < 1, x_n>0\right\}$ is an open half-ball. Assume $u \in C^2 (\overline{U^+}$) is harmonic in $U^+$ with $u=0$ on $\partial U^+ \cap \{x_n=0\}$. ...
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1answer
164 views

poisson's equation with robin's boundary, boundary value problem

Consider the Poisson’s equation with Robin’s boundary conditions as follows \begin{array}{ll} −\Delta u = f, &\text{in $U$,}\\ \frac{\partial u}{\partial \nu}+u=g, &\text{on $\partial U$,} ...
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84 views

Laplace's equation in polar coords

Question: Suppose that the function u(r, $\phi$) satisfies Laplace’s equation for plane polar co-ordinates (r, $\phi$) i.e. $$ ∇^2u = \frac{1}{r} \frac{∂}{∂r}(\frac{r∂u}{∂r}) + ...
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135 views

Show whether $\log r$ has a conjugate harmonic function on $\mathbb{C} \setminus \{0\}$

Can someone help me understand this passage in a student-written wiki article? The question is whether $u(x+iy) = \log \sqrt{x^2+y^2}$ has a conjugate harmonic function on $\mathbb{C}\setminus \{0\}$. ...
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1answer
63 views

$\nabla_y \left( \frac{1}{|x-y|} \right)= \frac{x-y}{|x-y|^3},$ right? Fundamental solution to Laplace in $\mathbb{R}^3$

OK, I can't figure out why I can't get this right: $$\nabla_y \left( \frac{1}{|x-y|} \right)= \frac{x-y}{|x-y|^3},$$ right? I've checked the calculation several times, although this student-written ...
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185 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 ...
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where $\nabla^2V = 0$ , evaluate $\int_S V d\Omega /4\pi$

Where $\nabla^2 V = 0$ in 3 dimensional Euclidean space, it is a well-known fact that $${\int_S V(\vec{r'}) d\Omega'\over 4\pi}=V(\vec{a})$$ where $\vec{a}$ is the center of a sphere $S$ of radius ...
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191 views

Poisson Equation in a Rectangle

The problem is to solve $$\Delta\phi=\frac \lambda {\varepsilon_0}\delta(x-x',y-y')\quad;\quad \phi(0,y)=\phi(a,y)=\phi(x,0)=\phi(x,b)=0.$$ My idea was to try and represent the RHS as a series of ...
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616 views

Show that the Poisson kernel is harmonic as a function in x over $B_1(0)\setminus\left\{0\right\}$

Show that the Poisson-kernel $$ P(x,\xi):=\frac{1-\lVert x\rVert^2}{\lVert x-\xi\rVert^n}\text{ for }x\in B_1(0)\subset\mathbb{R}^n, \xi\in S_1(0) $$ is harmonic as a function in $x$ on ...
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87 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 ...
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188 views

What is relationship between Wirtinger differential operator and multivariable chain rule?

What is relationship between Wirtinger differential operator(equation 5) and multivarible chain rule(equation 4)? for other Wirtinger related questions look here.
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2answers
365 views

Positive harmonic function on $\mathbb{R}^n$ is a constant?

Is it true that a positive harmonic function on $\mathbb{R}^n$ must be a constant? How might we show this? The mean value property seems not to be the way...for that we would need boundedness.
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1answer
61 views

square of a harmonic function bound

I need to solve this problem: Let $u$ be a harmonic function inside the open disk $K$ centered at the origin with radius $a$. We are also given that $\int_K u^2(x,y)dxdy=M<\infty.$ Show that ...