For questions regarding harmonic functions.

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Finding the harmonic conjugate

I have shown that the following function is harmonic and am attempting to find it's harmonic conjugate: $u=e^{-2xy}\sin(x^2-y^2)$ I know that to find the harmonic conjugate I need to use the ...
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55 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) ...
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118 views

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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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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116 views

is Poisson's kernel always integrable?

Let $E$ be a smooth domain. The Green function $G(x,y)=F(x,y)-\Phi(x,y)$ where $F$ is the fundamental solution to the Laplace equation and for fixed $x\in E$, $\Phi(x,\cdot)$ is a harmonic function in ...
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27 views

Dirichlet energy of solution to Laplace equation

Suppose $V\subseteq\mathbb{R}^3$ is compact with a smooth boundary. I'm interested in the Dirichlet problem $\Delta u=0$ subject to boundary conditions $u|_{\partial V}=f$ for a given function ...
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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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136 views

Prove or disprove: If $h$ is harmonic on $E$, then $h$ is constant on each $C_i$

For a general finite Markov chain $(X_n)_{n\in\mathbb{N}_0}$ with state space $E$ and transition matrix $P=(p_{x,y})_{x,y\in E}$, not necessarily irreducible, we define the linear space of ...
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Existence of a harmonic function?

I really have no clue about this problem. Should I represent $u(x)$ using $\varphi(x')$? But how to represent that? Thanks for any help!
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Subharmonic function and holomorphically parametrized integrals

Let $f_\lambda$ be a family of $L^1$ functions (say on $\mathbb{C}$) such that for all $z$ the map $\lambda \mapsto f_\lambda(z)$ is holomorphic. Consider the map $N(\lambda)=\log \int |f_\lambda(z)| ...
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The Schwarz reflection principle and harmonic function (Big Rudin chapter 11)

In his book page 250 Exer 11: Suppose that $I=[a,b]$,$\Omega$ is a region ,$I\subset\Omega$,$f$ is continuous in $\Omega$,and $f\in H(\Omega-I)$,prove that actually $f\in H(\Omega)$. If I follow the ...
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Geometric Interpretation of Laplace's Equation

Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be analytic. In the natural way, let $f = u + vi$ for $u,v : \mathbb{C} \rightarrow \mathbb{R}$. Let $z \in \mathbb{C}$. Suppose that $u$ and $v$ satisfy ...
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497 views

Poisson Equation in a Rectangle

The problem is to solve $$\Delta\phi=\frac \lambda {\varepsilon_0}\delta(x-x',y-y')\quad;\quad \phi(0,y)=\phi(a,y)=\phi(x,0)=\phi(x,b)=0.$$ My idea was to try and represent the RHS as a series of ...
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120 views

The second derivative of simple layer potential

A simple-layer potential is defined as $$\Psi(M)=\iint_{S}\dfrac{\sigma(N)}{R(M,N)}dS(N)$$ where $S$ denotes a flat region in the plane $z=0$; the coordinates of $M$ and $N$ are $(\rho,\phi,z)$ and ...
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Maps in $\mathbb{R}^n$ preserving harmonic functions

A map $\varphi:\mathbb{R}^n\to\mathbb{R}^n$ preserves harmonic functions if $f\circ\varphi$ is harmonic for every harmonic function $f:\mathbb{R}^n\to\mathbb{R}$. It is known that these maps are, in ...
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Superharmonic function and supermartingale

If $f$ is a nonnegative superharmonic function in dimension 2, how to prove that $f$ is constant? There is an exercise in R.Durrett's probability book, which gives out a method to prove it by ...
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$\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, $u:\overline{D} \to ...
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$G$-harmonic polynomials, dimension of $\text{Harm}(\mathbb{R}^n, S_n)$?

Definition. Let $\text{Harm}(\mathbb{R}^n, G)$ be the space of $G$-harmonic polynomials on $\mathbb{R}^n$. My question is, what is the dimension of $\text{Harm}(\mathbb{R}^n, S_n)$?
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Helmoltz equation on the torus

I am looking for the solution of the Helmoltz equation (or even Laplace, if not available) on the torus, that is, the manifold of line element \begin{equation} ds^2 = (c + a \sin(\theta))^2 ...
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Net flux zero equivalent to vanishing solution?

Consider \begin{align*} \Delta u(P) = 0, \quad P\in \Omega,\\ \frac{\partial u(P)}{\partial n_{p}} = f(P),\quad P\in \Gamma, \end{align*} where $\Omega$ is the infinite domain in the plane outside ...
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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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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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Harmonic function with boundary value 0 except one point

Let $\Delta=\{z\in\mathbb{C}: |z|<1\}$ Assume $u\in C(\overline{\Delta}\setminus \{1\})$ such that it is harmonic in $\Delta$ and $u(\xi)=0$ for $\xi\in S^1\setminus \{1\}$. (a) Find an example ...
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Difference of two subharmonic functions and signed measures

One of the reasons subharmonic functions are interesting is that if you take their laplacian, you get a measure (and conversely any finite Radon measure with compact support can be obtained in this ...
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Biharmonic boundary condition

I try to solve $$\Delta^2u=f$$ on unit square. with $f=4\sin(\pi x)\sin(\pi y)$ Using $v=-\Delta u,$ leads to $$v+\Delta u=0,$$ $$-\Delta v=f.$$ By Dirichlet boundary condition on $u$. What is ...
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Solutions of Laplace's Equation/Landau & Lifschitz Fluid Mechanics

in Fluid Mechanics by Landau and Lifschitz (a fairly well-known book - I'm just mentioning it for those who might have it) they are discussing "As we know, Laplace's equation has a solution l/r, ...
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subharmonic function and support functions

$M$ is a Riemannian manifold and $f$ is a continuous function on $M$. $f$ has the property that for any $p \in M,\epsilon>0$, we can find a smooth function $f_{\epsilon}$ such that ${f_\varepsilon ...
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358 views

Gradient of an harmonic function

Let $ M $ be a Riemannian manifold and let $ f $ be an harmonic function on $ M $. By Unique continuation theorem we can assert that if $ \nabla f = 0 $ on an open subset $ \Omega \subset M $ then ...
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expansion of function on spherical harmonics

Let $f :S^{n-1}\longrightarrow R_+$ be a square integrable function. Let $\{Y_{j,k}\}$ be an orthonormal basis of spherical harmonics on $S^{n-1}$, $j\in Z_+$ and $k=1, \ldots, d_n(j)$, where $d_n(j)$ ...
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Composition of a subharmonic function and a conformal mapping

this is q.4 of p.248 of Ahlfors book: Prove that a subharmonic function remains subharmonic after a composition with a conformal mapping. What I'v tried: Let $u:\Gamma \rightarrow \mathbb{R}$ and ...
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Biharmonic operator

Consider the problem: $$ \Delta^2 u = f$$ on the square domain $U=(0,1)\times(0,1)$ with boundary conditions: $$ u(x,y)=\Delta u(x,y) = 0$$ for $(x,y) \in \partial U.$ I try to solve it with the ...
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A function in the $L^2$ closure of the set of smooth, harmonic functions on the closed unit disk is smooth and has a harmonic representative.

This is is a homework problem I'm having trouble understanding. I am given the set $$Y=\{u\in \mathbb{C}^{\infty}(\bar{D_1})| \triangle u=0\}$$ and its closure with respect to the $L^2$ norm, ...
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Tight bounds for harmonic measure

I recently came across a question concerning harmonic measure here, and was wondering if there is a good reference summarizing different methods of estimating harmonic measure? Specifically, I would ...
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hyperbolic group ; showing the existence of a ration function with a certain condition

I'm currently working out of a book called differentialgeometry and minimal surfaces written by Jost-Hinrich Eschenburg und Jürgen Jost. Right now I'm looking at an exercise (12.5) under the ...
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453 views

Subharmonic/Superharmonic Inequality in Gilbarg/Trudinger [Section 2.8]

This is in Section 2.8 of Gilbarg and Trudinger. I believe there are some inaccuracies in the proof supplied, and in any case I think there is a more straightforward proof. Definition: A ...
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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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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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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 ...
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2D Laplace Equation with Sine-squared BC

I am having a bit of trouble solving the 2D Laplace Equation $$\nabla^2u(y,z) = 0$$ with 2 BCs being $\left.\dfrac{\partial u}{\partial y}\right|_{y=0} = 0$ $u\left(y=\frac{h}{2},z\right) = ...
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Real zeros of a function holomorphic in the upper half plane

Suppose that $f(z)$ is holomorphic in the upper half-plane and $\lim_{\, |z| \to \infty} f(z) = 1$ along any ray in the open upper half-plane. Suppose also that $f(z)$ has a continuous extension to ...
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Energy method for harmonic functions

I have two questions about the informations bellow that can be found in the book elliptic partial differential equations -QING HAN and FANGHUA LIN - chapter 1-pg 19 If $a_{ij} \in C(B_{1}(0))$ ...
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Show that the “Hartogs Regularity Radius” $R(z)$ is subharmonic

Exercise I'm a little stuck on an Exercise in Holomorphic Functions and Integral Representations in Several Complex Variables by R. Michael Range. The Exercise (E.II.5.1) is as follows: Let ...
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Harmonic non-negative function is constant

I'm having some trouble with the following: Let $u:\mathbb{R}^2\setminus\{0\}\rightarrow[0,\infty)$ be a harmonic function. Show that $u$ is constant. I have seen different proves for this. However, ...
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Neumann and Dirichlet Conditions for Schwarz-Christoffel Map

I'm looking to solve Laplace's equation on a polygon with Dirichlet and homogenous Neumann conditions using Schwarz-Christoffel (CS) mapping. I'm able to map the polygon to the upper-half plane using ...
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Laplace's equation in cylindrical coordinates for a function that only depends on the angle

I need to solve Laplace's equation: $$\nabla^2\Phi = 0$$ with the boundary conditions: $$\Phi(\theta=0)=0$$ $$\Phi(\theta=\pi)=a_1$$ In cylindrical coordinates ($r,\theta,z$), for $\Phi ...
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PDE: Laplace equation Maximum Principle

The maximum priciple for Laplace equation assumes, in both PDE textbooks by Fritz John and Lawrence Evans, that the domain of the harmonic solution be bounded. Is the maximum priciple still valid if ...
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Fourier transform on Laplace equations

We can solve some Laplace equations on the lectangular area by using Fourier transform. However many textbooks only introduce the cases when the area is given as a half plane or a strip. ; $y>0, ...
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Weighted Analogue of Mean Value Property

Let $f$ be a harmonic function defined on a planar disk $B \subset \mathbb{R}^2$ and continuous on $\partial B$. For which probability measures $\mu$ on $\partial B$ is it true that $$\int_{\partial B ...
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Numerical evaluation of the Laplace operator near the singularity points in spherical coordinates

If one had to evaluate $\Delta Y_l^m$ numerically everywhere on the unit sphere, including the singularity points $\theta = 0,\pi$, how would they do it? Let's say $Y_l^m$ is a spherical harmonic. I'm ...
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asymptotic of an interesting recurrence realtion (more general case)

A link to the original question for reference:Click here I tried to study a more general situation: Let $y_{d,d}=1$ and $$ ...