For questions regarding harmonic functions.

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Optimizing value of discrete harmonic function at a given point

Let $n>0$, and let $S_n$ denote the discrete square $S_n=[|-n,n|]^2$ (so $S_n$ has $(2n+1)^2$ elements). Let $K_n$ denote the set of four corner points $\lbrace (\pm n,\pm n)\rbrace$, and $C_n=S_n\...
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164 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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2k views

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 Cauchy–...
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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) ...
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65 views

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 (here $\...
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76 views

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 $u=...
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121 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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523 views

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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114 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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142 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 harmonic ...
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41 views

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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91 views

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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80 views

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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249 views

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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558 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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137 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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96 views

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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biharmonic complex analysis

Complex analysis texts typically discuss analytic functions whose real and imaginary components are harmonic and satisfy the Laplace equation, $\nabla^2 f = 0$. I am working with a complex function ...
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43 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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$\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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20 views

$G$-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 d\varphi^...
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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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38 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 way:...
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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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418 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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183 views

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 $C:\...
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215 views

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, $\bar{Y}$...
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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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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 $C^0(\...
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Trobule with an application of Green's representation formula

The teacher solved an exercise in class which required you to prove that, if $\Omega$ is a bounded domain in $\mathbb R^n$ and $G$ its Green function, then $G$ is symmetric, i.e. $G(x,y)=G(y,x)$ for ...
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Green's function for the Laplace or Helmholtz equations for an open rectangle in 2d or cylinder 3d?

I have only ever worked with free space Green's functions, or Green's functions for for the upper half space in 2d. So is it possible to determine a Green's function for the Helmholtz equation or ...
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Finite dimensional irreducible representations of Sp(2).

I am looking for an explicit description of the finite dimensional irreducible representations of the classical Lie group $\text{Sp}(2) = \{A\in M_2(\mathbb{H})\,|\,A\overline{A}^T = I\}$. I can ...
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Mean Value Property for harmonic functions

I would appreciate any insights on this matter; Let us consider the mean value property for harmonic functions in three dimensional euclidean space. This suggests that the value of a harmonic ...
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Harmonic Function - Multivariable calculus

One more exercise I stepped at while strolling through papers and journals for my preparation on the semester exams for multivariable calculus. Let $D=\{(x,y): x^2 + y^2 \leq 1\}$ A function $f:D \to ...
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Is there a mistake in this proof in Rudin's RCA?

Rudin's Real and Complex Analysis, 3rd edition, page 236 : How did Rudin get to $$ \frac{R-r}{R+r}u_n(a) \leq u_n(a+re^{i\theta}) \leq \frac{R+r}{R-r}u_n(a).$$ It seems to me that he used the mean ...
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Can the rank of harmonic map decrease far from the boundary?

This question is in some sense a continuation of this question, though it asks something weaker. Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we are ...
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Harmonicity on semisimple groups

Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in $U(\mathfrak{g})$ and let $f$ be a smooth (or analytic) real-valued ...
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What does it mean for a complex-valued function to be bounded above (or below)?

I was reading about the maximum-minimum principle for harmonic functions in my lecture notes, and it was formulated like this: Let $\phi$ be harmonic in a simply-connected domain $D$. If $\phi$ is ...
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3 dimensional harmonic conjugates?

An $n$ dimensional harmonic function is defined to be a real valued function $f$ in $\mathbb{R}^n$ such that $\nabla^2 f = 0 $. Equivalently, $f$ is the scalar potential of a conservative vector field ...
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80 views

Deduce the definition of a harmonic function in the context of a Markov Chain

We know from PDE that a harmonic function $f$ satisfies the mean value property, namely, $f(x)$ = $\frac{1}{\vert{B_r(x)}\vert}\int_{B_r(x)}f(y)dy$ where $B_r(x)$ is the ball about $x$ with radius $r$....