For questions regarding harmonic functions.

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Prove one property of harmonic function

Let $u(x)$ be a harmonic function defined in the square $[0,1]\times[0,1]$. Suppose that $u(x_k)=0$, where $x_k=(1/k,1/k).$ Prove that $u(x)=0$ everywhere in $[0,1]\times[0,1]$.
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35 views

Surface Integral of the Partial Derivative of a Harmonic Function

Assume that $V$ is a solid in $\mathbb{R}^3$ which is bounded by a surface $S$ whose normal is $\overrightarrow{n}$ and $f:V \rightarrow \mathbb{R}^3$ is a harmonic function on $V$. Show that ...
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1answer
59 views

Link between harmonic and holomorphic functions on a non-simply connected domain.

There is a theorem that states that if a function $h$ is harmonic on a simply connected domain, there exists a holomorphic function $f$ such that $h = Re f$. Now, I am having a problem with the ...
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1answer
37 views

Intuition behind estimates on derivatives of a harmonic function

In Evans' PDE book he gives the following theorem. Assume $u$ is harmonic in $U$. Then, $$ |D^{\alpha}u(x_0) | \le \frac{C_k}{r^{n+k}}||u||_{L^1(B(x_0,r))}$$ When asking my professor for some ...
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1answer
32 views

The expansion of harmonic function at infinity

If $u$ is a harmonic function on $\mathbb R^n$ outside some compact set such that $u$ goes to $1$ at infinity. Then does $u$ have the following expansion $$ u=1+\frac{a}{|x|^{n-2}}+O(|x|^{1-n})\quad ? ...
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74 views

Showing that $P_r(x)=\frac{1-r^2}{1-2r\cos x+r^2}\rightarrow 0$ uniformly on $[-\pi,-\delta]\cup[\delta,\pi]$ as $r\uparrow 1$

Let $0<r<1$ and consider the series $$s = \sum_{n=-\infty}^\infty r^{|n|}e^{inx}.$$ I have shown that the series converges uniformely to $$P_r(x)=\frac{1-r^2}{1-2r\cos x+r^2}$$ on all of ...
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66 views

What are the solutions of the Laplace equations for two (or more) eccentric cylinders?

I am looking for solutions to Laplace equation for two eccentric cylinders in 3D with arbitrary boundary conditions. The boundary condtions also depend on the axial variable. I tried to work with ...
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1answer
41 views

No Generalization of Mean Value Property for harmonic functions?

The Mean Value Property for harmonic functions tells us that the value of a harmonic function evaluated at the center of $D(P,r)$ equals its weighted integral over $\partial D(P,r)$. I am wondering if ...
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2answers
33 views

Find all harmonc radial functions.

Find all harmonc functions in C \ {0} wchich are constant on the circles $$ \{ z \in\mathbb{C} : |z| = r \} $$ How to start finding this functions?
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1answer
47 views

If the integrals of a harmonic function over horizontal lines are uniformly bounded, it is identically zero

Let $u\colon\mathbb{R}^2\rightarrow\mathbb{R}$ be a harmonic function, such that $$\int\limits_{-\infty}^{+\infty} \lvert u(x,y)\rvert dx < C,$$ where $C>0$ is a constant not depending on ...
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1answer
27 views

Linear span of poisson kernels dense in $L^1(\mathbb{T})$

A paper I am reading ("Schur's Algorithm, Orthogonal Polynomials, and Convergence of Wall's Continued Fractions in $L^2(\mathbb{T})$" by Sergei Khrushchev...really a great paper) repeatedly mentions ...
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1answer
136 views

Proof of uniqueness for the Poisson equation

Show that the following problem has at most one solution: Given a continuous function $\rho(x,y,z)$ which is zero for $x^2+y^2+z^2>a^2>0$, find $\phi$ such that $$\nabla^2\phi=\rho$$ ...
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2answers
85 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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1answer
63 views

Solution of the Laplace equation in polar coordinate.

Solve the following PDE: $$\phi(r,\theta) = \begin{cases} \Delta \phi=0 & \quad \text{for $a \le r\le b$ }\\[8pt] \phi=V & \quad \text{for $r=b$} \\[8pt] \phi+ C \sin(n\theta)=0 & ...
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2answers
47 views

Find a harmonic function in the cylindrical shell between $r=a$ and $r=b$

Calculate $\phi$, satisfying $\nabla^2 \phi=0$ between the two cylinders $r=a$, on which $\phi=0$, and $r=b>a$, on which $\phi=V$. I calculate it and found the solution is ...
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1answer
97 views

Solving the Laplace equation in a rectangle, using the separation of variables

Suppose I have $f_{xx}+f_{yy}=0$ on a region $R=\{(x,y):0\leq x\leq\alpha,0\leq y\leq\beta\}$ with boundary conditions $f(0,y)=f(\alpha,y)=0$, $f(x,0)=g(x)$, and $f(x,\beta)=h(x)$. I considered a ...
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39 views

harmonic function. How to prove?

I've with prove if $1 \over |x|$ is a harmonic function. I know with for a harmonic function, $f_{xx}+f_{yy}=0$, but I don't know how to derivate ${1 \over |x|} dx$. And I don't know how to derivate ...
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0answers
71 views

Construct holomorphic function from harmonic function

Let $h$ be a real valued harmonic function on the twice punctured plane $Ω=\Bbb C \setminus \{0, 1\}$. Show that there exist unique real numbers $a_0, a_1$ such that $$u(z)=h(z)−a_0 \log |z|−a_1 \log ...
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2answers
141 views

Laplace Equation on the Corners and Boundary of a Rectangle?

Consider for some rectangle $[a,b] \times [c,d] \in \mathbb{R}^2$, we have a generic boundary value problem: \begin{equation*} \begin{cases} \frac{\partial ^2 u}{\partial x ^2}+\frac{\partial ^2 ...
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135 views

the real part of a holomorphic function on C \ {0, 1}

Let $h$ be a real valued harmonic function on the twice punctured plane $Ω = \text{C \ {0, 1}}$. Show that there exist unique real numbers $a_0$, $a_1$ such that $u(z) = h(z) − a_0 \log |z| − a_1 \log ...
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79 views

Does anybody know how to actually derive spherical harmonics in a way that is historically accurate and intuitive?

And by "historically accurate", I mean without resorting to techniques of derivation which were developed after the fact or explanations which use the very concept they're trying to explain. The few ...
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45 views

Find a complex-differentiable function with real part $x^2(ay+8) +4y^2(y+b)$

Find a complex-differentiable function $f$ with real part $u(x,y) = x^2(ay+8) +4y^2(y+b)$ I have tried to use Cauchy-Riemann to get $v(x,y)$ but realised that I need to find the constants $a$ and ...
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1answer
46 views

Average Property of Harmonic Function

When we prove the average property of harmonic function, we use a formula \begin{align} & \int_{B_r(x)}\triangle u\,dy=\int_{B_r(x)}\text{div}(\triangledown u)\,dy \\[6pt] = {} & ...
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1answer
198 views

Zeros of a harmonic function

Prove that the zeroes of a Harmonic function is never isolated. All I can think of is a very rough idea of a proof by contradiction.
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1answer
50 views

Gradient of Harmonic Function

Theorem If $u\in C(\overline{B_R(x_0)})$ and is harmonic in $B_R(x_0)$, then $$|D^mu(x_0)|\leq\frac{n^m\exp(m-1)m!}{R^m}\max_\limits{\overline{B_R~(x_0)}}|u|$$ We can prove the theorem by induction, ...
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2answers
69 views

Can Laplace's equation be solved in a domain that is not simply connected?

I have a problem where the domain is like a box with a tube missing - e.g. 0< x<1,0< y<1, 0< z<1 less the region (x-0.5)^2+(y-0.5)^2 < 0.25 In order to solve Laplace's equation ...
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1answer
91 views

Prove the Inverse of a Nonconstant Harmonic Function is Unbounded

Let $u$ be a nonconstant harmonic function on $\mathbb C$. Show that for any $c\in\mathbb R, u^{-1}(c)$ is unbounded. Hint: $\{|z|>R\}$ is connected for any $R>0$. It seems like this proof ...
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1answer
152 views

Subharmonic function equivalent non-negative laplacian

I want to ask for a proof that if $v(x,y)$ is $C^2$ and is subharmonic [here, define as satisfyingthen $\Delta v \geq 0$ where $\Delta v = \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 ...
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1answer
82 views

Compact set of measure zero and sequence of Harmonic Functions with nice properties.

I was studying John B. Garnett's book Bounded Analytic Functions, and then I decided to try the following problem: Let $E \subset \mathbb{R}$ be a compact set, with $|E|=0$. Prove that there ...
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69 views

How to check the barrier function is superharmonic?

Suppose $n\geq 3$ and $\Omega$ is a bounded domain. In the Perron's method to solve the PDE \begin{equation} -\Delta u = 0 \text{ in } \Omega \quad \text{and } u = g \text{ on }\partial\Omega, ...
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34 views

Fourier Transforms of hyperspherical harmonics

I am trying to compute the Fourier Transform of a function on a 3-sphere, $f(\hat{Q})$, where $\hat{Q}$ is a unit vector in four-dimensional space. The function $f(\hat{Q})$ is expressed as a series ...
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1answer
110 views

How can I prove that $\partial\varphi\neq0$ implies $\bar\partial\partial\varphi>0$?

Let $\Omega\subseteq\Bbb C$ open and $\varphi:\Omega\to\Bbb R$ strongly subharmonic, $\varphi\in\mathcal{C}^2$ such that $\partial\varphi\neq0$. My problem is to prove that ...
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48 views

Show an equation only has harmonic solution

I want to show $$\begin{cases} \Delta(\Delta u) - \nabla\cdot (\Delta u \cdot \nabla u)=0\\ \int \Delta u < \infty\\ \Delta u \ge0 \end{cases}$$ in $\mathbb{R}^2$ only has a solution such that ...
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Are 1-D line sections of 2-D point source-invoked potential distributions positive definite?

Consider the 2-D potential distribution induced on the plane $y=0$ by a point source positioned at $(0, -y_0, 0)$ in the open halfspace below that plane. The material below the plane is assumed ...
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1answer
148 views

Source of the “$\cosh$ trick” for Laplacian eigenfunctions or Helmholtz equation solutions?

Suppose a smooth function $f : \mathbb{R}^n \to \mathbb{R}$ satisfies the Helmholtz equation, the PDE $\Delta f + k^2 f = 0$. A while ago someone showed me a trick: Define a function ...
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47 views

Must a function hold true for all (x,y) to be harmonic?

I've found lots of examples that show various functions that are harmonic but I still can't figure something out. Does a function have to hold true for all (x,y) to be considered harmonic or is it ...
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By direct computation, check that the function u(x) = $|x|^{1/2}$ cos($\theta$/2) is harmonic in the upper half plane H := {x = (x1, x2) | x2 > 0}.

Okay so Partial Differential Equations make no sense to me. Don't know what to do. All I know is that if a function u is harmonic then $\Delta$u = 0
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1answer
75 views

Determining if a Continuous $u:\mathbb{C}\to \mathbb{R}$ Satisfying some Property is Harmonic

If $u : \mathbb{C} \to \mathbb{R}$ satisfies $$u(x + iy) =\frac{1}{4}[u(x + a + iy) + u(x − a + iy) + u(x + i(y + a)) + u(x + i(y − a))]\tag{$*$}$$ for all $a$ then determine whether $u$ is harmonic, ...
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34 views

Harmonic solutions

Assume that $\Omega\subset R^2$ is an open bounded set with a smooth boundary, $g:\partial\Omega\to R$ is a continuous map and $\{b_i \ | \ i=1,2,\ldots,d\}$ is a finite subset of $\Omega$. ...
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Inverse Bessel Process as strict local martingale without Ito's formula?

Is there a way to prove that the inverse Bessel process $|B_t|^{-1}$ is a local martingale without using Ito's formula, considering that the Green's function $$g(x)=\int_0^\infty ...
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1answer
51 views

Describing Polynomials with Real Coefficients that are the Real Parts of Analytic Functions on $\mathbb{C}$

Describe those polynomials $$P=a + bx + cy + dx^2 + exy + fy^2$$ with real coefficients that are the real parts of analytic functions on $\mathbb{C}$. Idea (1): We are given that $P$ is the real part ...
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Maximum principle of harmonic function on compact manifold

Thm . (Maximum Principle) Let h be a harmonic function on a domain D in C . (a) If h attains a local maximum in D then h is constant. (b) Suppose that D is bounded and h extends continuously to the ...
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125 views

Reducing the Laplace equation with inhomogeneous BC's to the Poisson equation with homogeneous BC's

Given a domain $\Omega \subset \mathbb{R}^2$, one can reduce the Laplace equation $$\Delta u = 0, \qquad u = f \text{ on } \partial \Omega$$ to a Poisson equation $$\Delta v = g, \qquad v = 0 \text{ ...
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1answer
87 views

Finding the harmonic conjugate of $u(x,y)=\sinh(x) \sin(y)$

I know this is already a harmonic function but I am having trouble finding its harmonic conjugate. My instructor did this: $v_{x}=\cosh x \sin y \implies v(x,y)=\sin y \sinh(x)+g_{1}(x)$ ...
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1answer
40 views

2D Taylor expansion of F(x,y) where F(x,y) is harmonic (a solution of Laplace equation)

I would like to know if, given F(x,y) a real function of 2 variables that obeys $\nabla^{2} \left(F \left( x,y \right)\right) = 0$ , is it true that F(x,y) always equals its Taylor expansion within ...
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1answer
137 views

oblique derivative smoothness of harmonic functions

Let $Q$ be a domain in the half-space $\mathbb R^n\cap\{x_n>0\}$ and part of its boundary is a domain $S$ on the hyperplane $x_n=0$. Let $u\in C(\bar Q)\cap C^2( Q)$ satisfy $\Delta u=0$ in $Q$ and ...
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1answer
45 views

Is $\|x\|^6 \sin^6 \|x\|^6$ harmonic?

Suppose the function $$ u(x)=\|x\|^6 \sin^6 \|x\|^6$$ for $x \in \mathbb{R}^d$, where $$\|x\| = \sqrt{x_1^2 + \ldots +x_d^2}.$$ How can I decide if the function $u$ is harmonic in the unit ball ...
2
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1answer
118 views

Conformal transformation of complement of disk in upper half plane

Let $U$ be the complement in the half-plane $\operatorname{Im} z > 0$ of a disk of radius $a<1$ centered at $i$. I am looking for a conformal transformation that maps $U$ onto an annulus. Since ...
2
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2answers
145 views

Divergence structure equation

Consider Laplace's equation with potential function $c$: $$-\Delta u + cu = 0, \tag{$*$}$$ and the divergence structure equation $$-\operatorname{div}(aDv)=0, \tag{$**$}$$ where the function $a$ is ...
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47 views

Show that if $\lvert a \rvert \neq 1$, then the equation $\overline{z}^2 = az^2+bz+c$ has only a discrete number of solutions.

I knew the proof for this at some point, but I'm having trouble piecing it back together. At least, I think the proof I'm thinking of was for this result, or a result which implied this result. The ...