For questions regarding harmonic functions.

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1answer
27 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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0answers
25 views

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
51 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
21 views

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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1answer
23 views

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
36 views

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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3answers
41 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
66 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
34 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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1answer
35 views

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
46 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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0answers
19 views

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
24 views

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
36 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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0answers
30 views

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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0answers
23 views

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
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
15 views

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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1answer
32 views

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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22 views

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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0answers
18 views

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
20 views

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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0answers
43 views

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
24 views

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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1answer
23 views

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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1answer
19 views

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
10 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
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0answers
22 views

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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1answer
42 views

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
25 views

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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0answers
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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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0answers
31 views

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
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
18 views

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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2answers
98 views

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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0answers
23 views

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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0answers
20 views

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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0answers
58 views

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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1answer
53 views

Energy integral is convex for non-uniform diffusion equation in $\Omega\subset\Bbb R^n$

I'm having trouble proving that a certain integral that is a function of time, is a convex function. Let $\Omega\subset \Bbb R^n$ be a bounded Lipschitz domain, and let $u: ...
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1answer
24 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 ...
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0answers
29 views

Uniform convergence of harmonic functions to $0$ on compact subsets

Let $D \subset \mathbb{C}$ be an open, connected set and let $\{ u_n \}$ be a sequence of harmonic functions with $u_n: D \longrightarrow (0, \infty)$. Show that if $u_n(z_0) \rightarrow 0$ for some ...
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1answer
37 views

Given the holomorphic maximum modulus principle, prove Hopf's lemma

To smooth out my lecture notes, I'm looking for a derivation of Hopf's lemma for harmonic functions $u \colon D \subset \mathbb{R}^2 \to \mathbb{R}$ from the maximum modulus principle (and mean value ...
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1answer
36 views

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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0answers
50 views

Representing a function as a Poisson Integral.

This is a question I came across in Ahlfors' book Complex Analysis. It is found on page 171 of the 3rd Edition, Exercise 2. "Prove that a function $T(z)$ which is harmonic and bounded in the upper ...
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1answer
30 views

Can we change the Laplace equation to the wave equation with a linear substitution

I would like to know, is it possible to make a linear change of dependent and independent variables such that Laplace's Equation $u_{xx}+u_{yy}$ transforms to the Wave Equation ...
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2answers
33 views

On the partial derivatives of a harmonic function

Well, here is the thing. We know that the laplacian operator commutes with any partial derivative of a function, if the function is smooth. We also know that a harmonic function is infinitely ...
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1answer
34 views

Integrating a Poisson kernel in $n$ dimensional unit sphere

Let \begin{equation*} P(x,y)=\frac{1}{\omega_n R} \frac{R^2-|x|^2}{|x-y|^n} \end{equation*} be a Poisson kernel where $x$, $y$ are in $R^n$, $|x|<R$, $|y|=R$, $\omega_n$ is area of n dimensional ...
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1answer
45 views

Does a weaker form of the mean value property already imply harmonicity for continuous functions?

If $u:\mathbb{C}\to \mathbb{R}$ is continuous and satisfies $u(z)=\frac{1}{2\pi}\int_0 ^{2\pi}u(z+\frac{e^{i\theta}}{n})d\theta$ for all $n\in \mathbb{N}$ and $z\in \mathbb{C}$, is $u$ harmonic? What ...
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1answer
35 views

Eigenfunctions of $-\Delta_{S^n}$

I read somewhere that the eigenvalues of the Laplacian $-\Delta_{S^n}$ on the sphere $S^n$ consist of $k^2 + (n - 1)k$, with the corresponding eigenspace $V_k$ consisting of homogeneous harmonic ...