For questions regarding harmonic functions.

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1answer
32 views

Is the product of two subharmonic function necessarily subharmonic?

Defin: A continuous function $\varphi :\mathbb{R} \to \mathbb{C}$ is subharmonic if and only if, for any closed disc in $U$ with centre $\lambda_0$ and radius $r$,$$\varphi ({\lambda _0}) \le \frac{1}...
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0answers
19 views

Prove $\lim |u(x,y)|e^{-|x|}=0$ for harmonic function $u$

Let $\epsilon >0$ and $\Omega \subset \{(x,y): -\frac{\pi}{2}+\epsilon<y<\frac{\pi}{2}-\epsilon\}$. Let $u$ be harmonic function in $\Omega$ such that $u=0$ on $\partial \Omega$. Prove that ...
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0answers
85 views

Summation of series involving $\sinh$ of a square root

In a physics problem that I assigned in one of my classes recently, the following series arises: $$ S = \sum_{\text{odd } n} \sum_{\text{odd } m} \frac{(-1)^{(n+m)/2}}{nm} \frac{\sinh( \pi \sqrt{n^2 + ...
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1answer
46 views

Square integrable harmonic function

Let $u$ be harmonic function on $\mathbb{R}^n$ such that $\int_{\mathbb{R}^n} u^2 dx1...dxn < +\infty$ (1). I've got to prove that it implies $u=0$. My idea was to prove that (1) implies that $u$ ...
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0answers
20 views

Co-efficient Problem For Univalent harmonic functions on Unit disk

The Clunie Sheil Small conjecture for the second co-efficient of a univalent harmonic function on the unit disk is as follows:- Suppose, $h(z)+\overline{g(z)}$ is a one-one harmonic function on the ...
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0answers
54 views

Generalized Coupon Collector's Problem [duplicate]

I just read about the coupon collector's problem where you're trying to find out how many coupons you need to collect on average before you get one complete set. This turns out to be $nH(n)$. I was ...
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2answers
35 views

Derive mean value property of harmonic function using this method

We can view harmonic function $u(x,y,z)$ as a wave function that does not depend on time $t$. Let $\overline{u}(r)$ be its average value on the circle with radius $r$. Assume we have proved that it ...
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1answer
33 views

Laplace equation for a lens-shaped volume - any approximate analytical solutions?

I need to solve Laplace equation for a lens (drop on a surface) with constant boundary condition on the top surface and zero on the bottom surface (this is the simplest case, not the only one I ...
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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 ...
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0answers
50 views

Why does solving the Laplace equation on a graph using the method of relaxation get different results than a spectral method?

When I use the method of relaxation to solve Laplace's equation on a graph where the boundary conditions are fixing a set of nodes to either 0 or 1 then the solution ends up being entirely between 0 ...
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0answers
15 views

Surface Integral of a harmonic function and mean value property

I want to find a general expression of the following integral, where $h$ is a harmonic function (we're in $\mathbb{R}^2$): $\int_{|x-y|\leq a^2}\frac{h(y)}{\sqrt{a^2-|x-y|^2}}dy$ I think I can ...
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0answers
21 views

Harmonic function on 3-dimensional lamplighter group

Can anyone give an example of a non-constant bounded harmonic function on the Lamplighter group $\mathbb{Z}_2 \wr \ \mathbb{Z}^3$? This function should exist, because the random walk on this group has ...
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0answers
25 views

Fourier transform on forced linear harmonic oscillator

I'm having trouble solving the following equation: $\ddot x + \omega^2 x = a \sin \Omega t$, where $t >0$, $\omega \neq \Omega$, $x(0+) = 0 = \dot x(0+)$. It is asked to be done by Fourier ...
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33 views

Function $\phi (x,y) \mapsto \phi (u,v)$ [ie:…] under the transformation $f(z) = u(x,y) + v(x,y)$ where $f(z)$ is analytic & $d_z \,f(z)$ is not $0$

I am just having a bit of trouble understanding what I am being asked. The ie: in title says $\phi [x(u,v), y(u,v)]$ Part a) is asking to do the laplacian, $\nabla^2_{x,y} \phi (x,y)$ and to write ...
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1answer
51 views

Find a function $g(x,y)$ harmonic on $\{ 1<x^2+y^2<16\}$ such that…

In reviewing complex analysis, I stumbled upon the following problem: Find a function $g(x,y)$ harmonic on $\{ 1<x^2+y^2<16\}$ such that $g(x,y)=3$ when $x^2+y^2=1$ and $g(x,y)=8$ when $x^2+...
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1answer
41 views

Variational formulation of harmonicity on Riemannian manifolds

$\newcommand{\R}{\mathbb{R}}$ I am trying to follow a derivation of the first variation formula for the energy functional. (In "Selected Topics in Harmonic maps"). Here is the context: $M,N$ are ...
2
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1answer
26 views

Isometries of Riemannian manifolds are harmonic?

Let $(M,g),(N,h)$ be two Riemannian manifolds. Assume $f:M \to N$ is an isometric immersion. Is $f$ harmonic? (i.e a critical point of the energy functional) (I know this is true when $(N,h)=(\...
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2answers
21 views

Is this function harmonic?

$$u(x,y)=\frac{x}{x^2+y^2}$$ I'm trying to figure out if this function is harmonic. I worked out all the algebra and my conclusion is no because the numerator of the final fraction is $2x^3+2xy^2$ ...
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0answers
50 views

How to classify harmonic functions on the punctured disk without the Schwartz reflection principle?

I am working through old qual problems at the University of Minnesota and am trying to find an alternate solution to the following problem. Determine all continuous functions on $\{ z : 0 < \left| ...
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1answer
33 views

Partial derivative of x - is quotient rule necessary?

Let $$u(x,y)=\frac{x}{x^2+y^2}$$ I'm trying to determine if the given function is harmonic. I know that the 2nd partial derivative with respect to $x$ should, when added to the 2nd partial ...
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0answers
27 views

Find the harmonic conjugate of the following

I just want to make sure my reasoning is correct. I followed another similar question from this site. Also, is there a method I can use after coming to the conclusion below to check to make sure my ...
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0answers
35 views

How do I see if $g$ is a polynomial or not??

Let $u$ be a real valued harmonic function on $\mathbb{C}$. Let $g: \mathbb{R^2} \to \mathbb{R}$ be defined by: $$g(x,y)=\int_0^{2\pi}u(e^{i\theta}(x+iy))\sin \theta \,d\theta$$ Which of the ...
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0answers
59 views

Harmonic map into $S^n \times \mathbb{R}$

Consider a harmonic map $\Phi : \Sigma \to S^n \times \mathbb{R}$, where $\Sigma$ is a surface, and the metric on $S^n \times \mathbb{R}$ is given by the product metric. Choose local spherical (polar) ...
3
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1answer
41 views

A version of Casorati-Weierstrass for harmonic functions?

Suppose that $f:B(0,1)\setminus\left\{0\right\}\subset \mathbb{R}^n \to \mathbb{R}$ is a harmonic function. Clearly, the property that $\overline{f(B(0,\epsilon))}=\mathbb{R}$ for all $\epsilon>0$ ...
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0answers
56 views

Harmonic map and pullback metric

Let $\phi : M \to \mathbb{R}^n$ be a harmonic map, where $M$ is a Riemannian manifold. Let us take coordinates $(u_1, u_2,..., u_n)$ on $\mathbb{R}^n$ and express the Euclidean metric as $g = \Sigma ...
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2answers
26 views

Harmonic Motion - Fourier Approximation What does this mean below?

There is a method to solve systems under harmonic loading, harmonic balance method, which is basically obtaining fourier expansions of unknown response quantities and solving for coefficients of ...
3
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1answer
70 views

Interior estimate for derivatives of harmonic function

I'm learning PDE from the book of Trudinger and Gilbarg and I'm attempting to prove the following theorem: Let $\hspace{0.1ex}u\hspace{0.1ex}$ be harmonic in $\hspace{0.1ex}\varOmega \subset \...
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1answer
38 views

Harmonic Conjugate (Multiplication, power, addition/subtraction)

The Question Suppose that $v$ is a harmonic conjugate for $u$ on a domain $D$. Prove that $u(x,y)^3 - 3u(x,y)v(x,y)^2$ is harmonic. I'm trying to prove that this function is also harmonic when $v$ ...
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0answers
29 views

Value of $f_x g_x+ f_y g_y + f_z g_z$ when $f$ and $g$ are harmonic

Let $f$ and $g$ be distinct real-valued harmonic functions, which not merely differ by a constant or are not merely multiples of each other. Also, assume that the first order partial derivatives of ...
3
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3answers
154 views

I need to solve $\phi (x,y) = \frac{2V}{\pi} \int_{0}^{\infty} \frac{\sin(kx)\cosh(ky) dk}{k\cosh(ka)}$

I start with a integral in complex plane $$\oint_c \frac{e^{izx} e^{zy} dz}{z\cosh(za)}$$ where $c$ is a countour starting in $z = -R$ along the real axis and jumping the pole at origin and continuing ...
0
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1answer
48 views

Laplace-Beltrami of the Gauss map

I'm looking for the proof of very nice identity about the Laplace-Beltrami operator of the Gauss map $N$ of a regular surface in $\mathbb{R}^3$ given by a patch $X$. I want to show that $$\Delta N = -...
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0answers
28 views

Laplacian on the sphere (proof of a proposition)

$$ Let \;f:U \to \mathbb{R}\;be\;a\;function\;on\;an\;open\;subset\;of\;S^{m-1}\;which\;is\;the\;restriction\;of\;a\;smooth\;function\;F:U{'} \to \mathbb{R}\;defined\;on\;an\;open\;subset\;of\;{{\...
0
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1answer
41 views

Find all the harmonics functions constants on the rays

I'm stuck with this exercise, I don't know how characterize the harmonic functions of the exercise. I'd appreciate your help. Thank you. Let $G=\mathbb C\setminus\{(-\infty,0]\}$. Find all the ...
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0answers
25 views

Non trivial boundaries for laplacian equation on rectangle

1) Can this Laplace equation, with its non trivial boundaries (on a rectangular domain), be solved analytically? $$\frac{d^2U}{dx^2}+\frac{d^2U}{dy^2}=0$$ $$U_x(0,y)=0\quad,\quad U_x(a,y)=f(y)$$ $$...
2
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1answer
55 views

Dirichlet problem to the ball with boundary data $1-2y^2$.

Let $\omega=\{(x,y):x^2+y^2<1\}$ be the open unit disk in $\mathbb R^2$ with the boundary $\delta\omega$.If $u(x,y)$ be the solution of Dirichlet problem $$\begin{cases} u_{xx}+ u_{yy}=0 & \...
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0answers
70 views

Whittakers general solution to Laplace and its relation to separable variables

So It is well known that the 2D solution to the Laplace equation can be obtained by changing to complex coordinates $u=x+iy$ and $v=x-iy$. This can be extended to n dimensions as long as the complex ...
1
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1answer
47 views

Harmonic function zeros on open subset

Let $h$ be an harmonic function on $\Omega\subset\mathbb{C}$. Let $A\neq\emptyset$ an open subset of $\Omega$ such that $h\mid_A\equiv 0$. Prove $h\mid_\Omega\equiv 0$. I thought on taking a point ...
0
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1answer
197 views

Prove the uniqueness of poisson equation with robin boundary condition

We have $\Delta u=f$ in $D$, and $\dfrac{\partial u}{\partial n}+au=h$ on boundary of D, where $D$ is a domain in three dimension and $a$ is a positive constant. $\dfrac{\partial u}{\partial n}=\...
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1answer
65 views

The average of a subharmonic function on a circle increases with radius

Let $u$ be a subharmonic on open set $\Omega$. Let $a\in\Omega,R>0$ such that $B(a,r)\subset \Omega$. Prove $$v(\rho)=\int_0^{2\pi}u(a+\rho e^{it})dt$$ is a monotone increasing function on $(0,R)$. ...
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3answers
68 views

Prove a function is harmonic

This problem is from Conformal Mapping by Zeev Nehari: If $u(x,y)$ is harmonic and $r=(x^2+y^2)^{1/2}$, prove $u(xr^{-2}, yr^{-2})$ is harmonic. The hint is obvious: "Use polar coordinates." I ...
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1answer
37 views

Extending a harmonic function

Suppose that $u$ is a harmonic function on some open set $U$ (assume that $\overline{U}$ is compact). It is well known than in this case $u$ is smooth. Is it true that we can extend $u$ to the whole ...
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0answers
22 views

How to generalize a fact (convex function of a mtg is submtg) about martingales to multivalued martingales?

It's known that a convex function of a martingale is a submartingale. What about martingales with values in $\mathbb{R}^{n}$? Is is true that a subharmonic function of such a martingale is a ...
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2answers
42 views

Strong maxima and minima

I'm stuck with this problem, in particular at b): Let $u:D \subseteq \mathbb{R}^2 \to \mathbb{R} \in C^2$ a harmonic function. $u$ has a local maximum at point $\vec{p} \in D$. Then: (a) Show that, ...
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0answers
36 views

$\nabla^2 u = 0 $ and integral $u$ around $\partial B_\rho$

I'm doing exercises from book about vector calculus. And there is problem wich I'm not sure what to do. Let's see the hypotesis. Let, $D \subseteq \mathbb{R}^2$ an open set. Let, $u:\overline{D} \to \...
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3answers
241 views

Is there a reason why Harmonic functions are defined on open sets?

Whenever I see a definition of a harmonic function, it's always defined as follows A function $f : U \to \Bbb{R}$ is called harmonic (where $U$ is an open subset of $\Bbb{R}^n$) iff it is twice ...
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1answer
41 views

The averages of a subharmonic function over concentric balls increase with radius

Let $B_r$ a ball of radius $r$ in $\mathbb{R}^n$ and $u \in H^{1}(B_r)$ with $\Delta u =0$ in the weak sense. I am reading a paper and the author says that : "since $|\nabla u |^2$ is subharmonic, ...
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2answers
72 views

Differentiating this integral,

I want to show that $u_{xx} + u_{yy} = 0$ for the integral given below, so I think I want to differentiate under the integral with respect to both $x$ and $y$. The goal is to show that $u$ is ...
3
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1answer
35 views

How to construct such a harmonic function on the upper half plane of $\mathbb{C}$ satisfying the following condition?

(1)Let u be a bounded harmonic function on the upper half plane of $\mathbb{C}$. Show that $\forall y$ we have $u(x+iy)=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{y\cdot u(t)}{(t-x)^2+y^2}dt$ for $x,y\...
1
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1answer
51 views

Some basic questions regarding the Maximum Principle for Harmonic Functions,

I've seen a uniqueness argument come up a few times but I don't really understand it. The argument is that if two harmonic functions $f$ and $g$ agree on the boundary of some domain, then $f-g$ or $g-...
0
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2answers
38 views

Find all possible functions, $F(r)$, harmonic in $2$ and $3$ dimensions,

Sources: this is an old advanced calculus exam question, which I think is asking for harmonic functions. The problem statement is: Suppose $F(r)$ is a smooth function of $r$ for $r>0$ . Define ...