For questions regarding harmonic functions.

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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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18 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
16 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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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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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
35 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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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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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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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
16 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
91 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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22 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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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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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
51 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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26 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
34 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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49 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
29 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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27 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
33 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 ...
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1answer
29 views

Establishing a Variant of the Mean Value Property of Harmonic Functions

Let $u:U\to \mathbb{C}$ be harmonic and $\overline{D}(P,r)\subset U$. Verify the following variant of the mean value property of harmonic functions: $$u(P)=\frac{1}{2\pi r}\int_{\partial ...
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1answer
43 views

laplacian of $1/\rho$ in cylindrical coordinates

In spherical coordinates, I believe that the laplacian of $1/r$ is zero everywhere except at $r = 0$ or \begin{align} \nabla^2 \dfrac{1}{r} = -4\pi \delta^{(3)}({\vec{r}}). \end{align} where $r$ is ...
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1answer
15 views

Showing a function is harmonic on a domain - Imaginary part of $(A\cosh(z)+\frac\pi z)$

How to know that $\text{Im}(A\cosh(z)+\frac\pi z)$ is harmonic on domain $\{z|0\lt\text{Im }z\lt \pi\}$ where $A\in\Bbb R$? I am not sure how I would verify Laplace's equation here(which I imagine is ...
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1answer
33 views

Harmonic Function with linear growth

We want to find harmonic functions $w$ in (say) $\mathbb{R}^2$ that are zero on $\{y=0\}$ with the linear growth bound \begin{equation} \sup_{\mathbb{B}_R} |w| \leq C(1+R) \end{equation} where ...
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1answer
32 views

Proving that $\inf_{\nu}\varphi_{\nu}$ is subharmonic

Let $\Omega\subseteq\Bbb C$ open, $\varphi:\Omega\to[-\infty,+\infty[$ is subharmonic on $\Omega$ if $\varphi$ is uppersemicontinous on $\Omega$ For all $K\Subset\Omega$ compact, and for all ...
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2answers
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An upper semicontinous function which is not subharmonic.

Let $\Delta\subset\Bbb C$ be the open unitary disk. Let $\varphi:\Delta\to\Bbb R$ defined as follows: $\varphi(z)=1$ if $\Re z\ge0$, $\varphi(z)=0$ otherwise. So $\varphi$ is upper semicontinous. In ...
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1answer
23 views

Probability denisity function of simple harmonic

Suppose that a spring is oscillating up and down with vertical position given by $u(t) = \sin(t)$. If you pick a large number of random $t$ to look at the position, then prove that the PDF is ...
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1answer
23 views

Schwarz Reflection Principle for Harmonic Functions

Given $\Omega \subset \mathbb{R}^n$ define $\Omega^+ = \Omega \cap \{x_n>0\}$ and $\Omega^0$, $\Omega^-$ analogously let $u \in C^2(\bar{\Omega}^+)$ be harmonic and such that $\frac{\partial ...
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28 views

Harmonic Functions on Connected Open Set

From the maximum principle, any unbounded harmonic function $u : \Omega\rightarrow \mathbb R$ on a connected open set must be surjective. If $\Omega$ is bounded, does there always exist such a $u$?
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28 views

Harmonic function with vanishing partial derivative

Let $u:D(0,1)\to \mathbf{R}$ be harmonic on the unit disc, and suppose there exists a $z_0\in D(0,1)$ such that all partial derivatives of $u$ vanish. Show that $u$ is constant. I found this problem ...
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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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1answer
26 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
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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
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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
20 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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66 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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0answers
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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
24 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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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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42 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
22 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
69 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
62 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 ...