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# Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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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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### Can the rank of harmonic maps decrease far from the boundary?

Let $\mathbb D^n$ be the closed unit disk in $\mathbb R^n$. Let $f:\mathbb D^n \to \mathbb{R}^n$ be a smooth immersion . ($df$ is everywhere invertible). Let $\omega:\mathbb D^n \to \mathbb{R}^n$ be ...
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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 Cauchy–...
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### All second partial derivatives of harmonic function are $0$

I am given this question as a homework assignment. Assume that $f$ is from $\mathbb R^2$ to $\mathbb R$ and has a strict local maximum at $(x_0, y_0)$. prove that all second partial derivatives of ...
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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 way:...
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### 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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### 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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### Charge distribution on an arbitrarily shaped conductor

From physics we know that given a charged conductor put in vacuum (no external electric fields), the charge distribution on its surface is approximately proportional to the curvature of the surface on ...
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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!
### Show that $\log \left| z \right|$ is harmonic and find its the conjugate harmonic function.
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)| ...