For questions regarding harmonic functions.

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Deriving the Equation for the Depth to Which an Object Falls

For a body of mass m kg, show that the depth to which the body would fall if attached to a rope, with a length of l meters. The depth is given by the model: $$d= \frac{2ml \pm l \cdot \sqrt{4m^2 + ...
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1answer
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Question about fundamental solution to the biharmonic operator $\Delta^2=\Delta(\Delta)$?

Show that when $n=2$, the function $u(x)=-\frac{1}{8\pi}|x|^2log|x|$ is a fundamental solution to the biharmonic operator $\Delta^2=\Delta(\Delta)$. That is, show that $$\varphi(0)=\int_{\Bbb ...
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Harmonic functions and null recurrence

Let $(X_n)_{n \geq 0}$ be a irreducible Markov chain defined on a countable state space $S$. It is known some ways to figure out if this chain is recurrent or not looking for superharmonic functions ...
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1answer
42 views

A question regarding harmonic function.

Can any one provide some hint on the following question? I have being thinking about this for a while but cannot figure out where to start. I have been thinking about Taylor expansion but it seems not ...
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1answer
53 views

Describing the zero level set of a harmonic function

Let $p(k)$ be a complex polynomial of degree $n\in\mathbb{N}$. Let $A=\{k\in\mathbb{C}:\text{Re}\,p(k)=0\}$ The harmonic function $\text{Re}\,p(k)$ determines the behaviour of $A$. Fix $z\in A$. If ...
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1answer
28 views

Bounded harmonic function on $\mathbb{R}^3$

Any suggestions how to get started? I know Liouville's theorem, but not sure how to apply it here: Let $u$ be a harmonic function on $\mathbb{R}^3$. Assume there exists $C>0$, independent of $x$, ...
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1answer
27 views

harmonic functions on the disk which agree on the real are identical?

The question is whether it is possible to find two distinct harmonic functions on the unit disk $\mathbb{D}=\{z: \ |z|<1 \}$ such that they agree on $\mathbb{D} \cap \mathbb{R}$. If yes, please ...
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Uniformly bounded family of harmonic functions

I am pretty sure other questions on this site can answer this problem, but I'm really interested in knowing if this particular solution is valid. Thanks. Question: Let $U$ consists of the set of ...
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1answer
50 views

How to numerically solve the Poisson equation given Neumann boundary conditions?

I want to solve the Poisson equation on a 2D domain given Neumann-type boundary conditions: The PDE: \begin{equation} \nabla^2 \; u(r,\theta) \;=\; f(r,\theta) \end{equation} The boundary ...
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1answer
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Is the harmonic function constant?

Suppose $f$ is harmonic on $\mathbb{R^{2}}$ and constant on a neighbourhood in $\mathbb{R^{2}}$. Is $f$ constant on $\mathbb{R^{2}}$?
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Question on real-valued harmonic function

Let $V\subset\mathbb{C}$ be a connected open set and $u$ a real-valued harmonic function on $V$. Suppose that the set $S=\{p\in V \mid \nabla u(p)=0\}$ has a limit point in $V$, then $u$ is constant. ...
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1answer
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What theorem is this? (in PDE)

I'm confused because it's titled as "Gauss's theorem about heat flux" (not in English though, I'm translating), but instead of the heat equation there's Laplace's equation written above the theorem. ...
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Construct holomorphic function from harmonic function

Let $h$ be a real valued harmonic function on the twice punctured plane $Ω=C$\ {0, 1}. Show that there exist unique real numbers $a_0, a_1$ such that $$u(z)=h(z)−a_0log|z|−a_1log|z−1|$$ is the real ...
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3answers
45 views

Is the boundedness necessary to extend harmonically?

"If $u$ is harmonic and bounded in the punctured disk $0<|z|< \rho$, then $u$ can be extended harmonically to the disk $|z|<\rho$ harmonically." This fact has been shown here. My Question ...
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1answer
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Uniqueness of harmonic function with Mixed Dirichlet Neumann condition

Let $u \colon \{\mbox{Im } z>0\}\subset\mathbb{C}\to \mathbb{R}$ be a positive harmonic function in the upper half plane, i.e $$ \Delta u=0,\,\, \mbox{for}\,\mbox{ Im } z>0. $$ Consider now the ...
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1answer
68 views

How to differntiate $\int_{0}^{2\pi} u(re^{i\theta}) d\theta$?

Suppose $u$ is a twice continuously differentiable function on $a< |z|<b, \ z\in \mathbb C,$ which is harmonic that is, it satisfies $u_{rr}+\frac{1}{r}u_r + u_{\theta \theta}=0.$ (If we put ...
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1answer
36 views

Prove that a harmonic function is an open map.

I'm trying to solve the following exercise of the book Functions of one complex variable, John B. Conway on page 255: 4. Prove that a harmonic function is an open map. (Hint: Use the fact that the ...
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Lemma in Farkas/Kra: Riemann surfaces on construction of domain satisfying certain properties

I'm having some trouble understanding the proof of this lemma. I can follow the construction of $u$ and $D$, however the final step in the proof seems to be without justification. Specifically why ...
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1answer
38 views

An application of strong maximum principle for harmonic functions

This should be an easy application, since it is given as a remark without proof in Evan's pde book. Let U be connected and u harmonic in U. Then u is positive everywhere in U if u is positive ...
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1answer
170 views

Topology of solution to a nonlinear eigenvalue problem

Consider the elliptic PDE: $$-\Delta u= f(x) u. $$ Assume that $f,u$ are defined in some reasonable bounded domain $\Omega \subset \mathbb{R}^n$ and impose the boundary condition $u=0$ on $\partial ...
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1answer
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Laplace equation in spherical coordinates

I am trying to calculate the Laplace equation($\Delta f =\partial_x\partial_xf + \partial_y \partial_y f + \partial_z \partial_z f = 0$ ) in $\Bbb{R}^3$ for spherical coordinates. $$g(r, ...
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1answer
108 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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Solution procedure for poisson equation

Consider the Poisson equation in the rectangle $Q=\{(x,y):0<x<1,0<y<1\}$, $$u_{xx}+u_{yy}=F(x,y)$$ $$u(0,y)=0,\,u(1,y)=0$$ $$u(x,0)=\phi(x),\,\,u(x,1)=\psi(x)$$ My Question: Is ...
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How to differentiate a harmonic function presented by Poisson integral formula

Let $h(x+iy)$ be a harmonic function in the open neighbourhood of the closed unit disc $\overline\Delta(0;1)$ of $\mathbb{C}.$ Then it can be presented by Poisson integral formula in the following ...
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1answer
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Fourier Transform of spherical harmonics

I am trying seeking for definition (or some source) of the Fourier Transform of Spherical Harmonics (see https://en.wikipedia.org/wiki/Spherical_harmonics). Any help will be really appreciated. ...
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1answer
29 views

Converse to mean value property: ball mean value property implies harmonicity

It is well-known that harmonic functions satisfy the mean value property. That is, if we set $\alpha(n)$ to be the volume of the unit $n$-ball, we have the following theorem. Let $u$ be an ...
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1answer
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Show that a nonconstant subharmonic function on a manifold cannot attain its supremum

PROBLEM: Suppose $f$ is a smooth non-constant function on a connected Riemann manifold $M$ of dimension 2 such that $f$ is bounded and $\Delta_M f \ge0$. Show $f$ cannot attain its supremum. I try ...
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1answer
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Maximum principle, quantitative version

Is it true that $$ \|f\|_{L^\infty(\Omega)}\leq C\|\Delta f\|_{L^\infty(\Omega)}+C\|f\|_{L^\infty(\partial\Omega)} $$ for all $f\in C^2(\Omega)$ and smooth $\Omega$?
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Reference For a PDE text that treats non homogenuous boundary conditions Rigorously

I am interested in reading a text or paper where elliptic and parabolic PDE's are discussed on bounded domains with non-homogeneous boundary conditions. I haven't been able to find anything in the ...
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2D Poisson equation and Bessel Functions

If I were to solve a inhomogeneous 2D Laplace's equation (in polar coordinates), of the form: $$\nabla^2 f= J_2(r)$$ How would I use the Hankel or the Mellin transform to find a solution? I couldn't ...
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2answers
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Problems identifying harmonic motion

Not sure why I am having so much trouble with this. I have a function f(t) = -cos(t) + 3sin(t-pi/6). I am trying to find the amplitude, period, and phase angle. But, I am under the impression that ...
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1answer
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Unicity of solution for a parabolic problem?

How can I show that the parabolic problem $$ \begin{cases} \partial_xu- \Delta u=0 & \mathbb{R} \times (0,+ \infty)\\ u(x,0)=f(x) & \mathbb{R} \end{cases} $$ has a unique solution? Can ...
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Biharmonic operator; properties, identities

The biharmonic operator is $\nabla^4 \phi \equiv \nabla^2 (\nabla^2 \phi)$. Are there any identities for it? I need to find $\phi$ such that $~\\$ $\nabla^4 \phi = \frac{1}{3}\nabla^4 u^3 - u ...
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1answer
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Laplace equation in a circle with non-continuous Dirichlet boundary conditions

I have to solve: $$ \begin{cases} u_{rr}+\frac{1}{r}u_{r}+\frac{1}{r^2}u_{\theta \theta}=0 & [0,1) \times [-\pi,\pi] \\ u(1,\theta)=0 & (-\pi,0) & (1) \\ u(1,\theta)=1 & (0,\pi) ...
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2answers
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Connection Between Convergence on Natural Boundary and Weierstrass Functions

So, I was fooling around thinking about constructing functions on the unit disc $\mathbb{D}$, which cannot be extended to the boundary by Hadamard's Gap Theorem. At first I constructed the function ...
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1answer
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non constant harmonic function

If $u$ is harmonic function on disk with radius $R$ around the origion, and non constant in it. why is it true that $u$ cannot be constant in any sub-Disk (i.e disk with radius less than $R$) thanks ...
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1answer
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How to solve 2D Laplace Equation over an infinite rectangular strip (bounded on two edges), with Dirichlet boundary conditions

Is it possible to solve Laplace equation $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}=0$, over an infinite rectangular strip defined by $0 < x < \infty$ and $0 < y ...
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1answer
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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
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1answer
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General solution of laplace equation

I want to know that: What is the general solution of the (2 and 3 dimentional)Laplace equation $f_{xx}+ f_{yy}=0$ and $f_{xx}+ f_{yy} +f_{zz}=0$? With many thanks for your help.
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Conformal transformation of a region bounded by a curve $y=x^a, a \in \mathbb{R}$

I would like to solve the 2D Laplace equation $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} =0$ on the positive upper half plane: $0 <x<\infty$ and $0 < y < ...
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about the mean value formula for harmonic functions

Let $ u \in C^{2}(B_R(x)) \cap C(\overline{B_R (x)}) $ a harmonic function. Does the function $u$ satisfies $$ u(x ) = \frac{1}{\omega_n R^{n-1}} \int_{\partial B_R (x)} u = \frac{1}{\omega_n ...
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Poisson's integral equation

Thank you. How can I find an harmonic function in the unit circle, that takes the value of \begin{equation*} F(\theta)= \left\lbrace \begin{array}{l} +T \text{ if } 0<\theta<\pi \\ ...
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1answer
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Prove a function is subharmonic?__Gilbarg Exercise

My attempt: By a long time google search I found in a book about how to prove it in the case of harmonic function. Just construct another function $v$ which is harmonic in the $\Omega$, then prove ...
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1answer
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Proof that Y=Acos(px)+Bsin(px) is only periodic if p=n

I am asked to show that a function y=Acos(px)+Bsin(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 for both ...
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building an orthogonal grid on 2d unbounded domain by solving laplace's equation using FEM

I am trying to build an orthogonal grid on an unbounded two dimensional domain by solving Laplace equation with FEM. As an example, consider the domain $D$ defined as $x \in [0,1], y\in[f(x),y_0]$ ...
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Question about an boundary integral equation with a jump in the boundary

I have the following problem: $$\Delta u = 0\;in\;\Omega$$ with several boundary conditions. Applying Green's second identity the representation formula can be derived: ...
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1answer
565 views

Complex and real forms of the Poisson integral formula

In my complex analysis book there is the expression $$\frac{1 - |z|^2}{|1 - \bar z e^{it}|^2}$$ and it says that when $z = re^{it}$, we can write the above expression as $$P_r(t) = \frac{1 - r^2}{1 - ...
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Constructing a theoretical solution to a non-homogeneous Dirichlet problem from known solutions

To begin, let $\Omega\subset\Bbb R^n$ be whatever kind of domain we like, and let $$\begin{align}f&:\Omega\times(0,+\infty)\to\Bbb R \\ d &:\partial\Omega\to\Bbb R \\ g&:\Omega\to\Bbb ...
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1answer
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Harmonic function in $\mathbb{R}^n$ is not one-to-one, for $n\geq 2.$

Le $u:\mathbb{R}^n\to\mathbb{R}$ a harmonic function. Prove that if $n\geq2$ then every $y\in Im\{u\}$ is attained infinite times, but it's not true for $n=1$. I no have idea to start, someone has a ...
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1answer
25 views

Maximum of a subharmonic function on it's boundary.

I am trying to solve below problem of the book Partial Differential Equations(Third edition) written by jurgen just, problem 3.9. Can any one give an idea? Thanks in advance. Let $\Bbb ...