# Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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### Laplace equation between circles with Dirichlet boundary condition

I am wondering if the below problem is solvable or not. Laplace equation between two circle with raduses equal to $r=1$ and $r=r_0$ where $r_0<1$: $\nabla^2 \phi(r,\theta)=0$ The boundary ...
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### Let $u$ be a harmonic function in $\mathbb{R}^n$ with $\int_\mathbb{R} |u|^p < \infty$. Then $u \equiv 0$.

This post Let $u$ harmonic. Then $\int_{\mathbb R^d}|u|^2<\infty \implies u=0$ answers this question in the case when $p=2$ using the Cauchy-Schwarz Inequality and an application of the Mean-Value ...
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### question on Poisson equation with solution not in $H_0^1$

I'm considering the following quesion about Poisson equation: $$-\Delta u=f$$ in a ball radius $1$ in $3$ dimension, if $f\in L^{2}$, then the theory of elliptic PDE says that the above equation ...
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### Math discrete: Proof by induction of harmonic serie

I need to prove by induction the following. The left hand side of the inequality is basically a harmonic serie where $H(1065^k)$. I've been able to complete this problem with $H(2^k)\ge 1 + \frac k2$, ...
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### Comparison principle

This principle is from the harmonic function theory. Let $\Omega \subset \mathbb{R}^d$ be a bounded, open subset of $\mathbb{R}^d$ with compact closure $\overline{\Omega}$ for some $d \in \mathbb{N}$ ...
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### Sum of squares of harmonic functions

I have to show $w(x)=\sum_ju_j^2(x)$ has its maximum value on the boundary $\partial\Omega$. The functions $u_j$'s are all harmonic. I think I need to show that $w$ is also harmonic and then I can use ...
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### Weak Maximum Principle as a corollary of Strong Maximum Principle?

I am reading http://www.axler.net/HFT.pdf. The author proves the following : $\Omega$ is an open connected set and $u$ is real-valued and harmonic in $\Omega$. If $u$ attains a maximum in $\Omega$ ...
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### Showing that Green's function on a domain can be expressed in terms of Green's function on a conformally equivalent domain

First, this is the definition I have for Green's function on a simply connected domain: let $\Omega$ be a simply connected bounded open set and let $z_0 \in \Omega$. Then Green's function on $\Omega$ ...
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### An easy way to solve this Poisson equation?

Problem : $-\Delta u=1-\sqrt{x^2-y^2}$ in $B_1(0)$; $u=0$ on $\partial B_1(0)$ I could use the Poisson integral formula : $$u(x)=\int\limits_{B_1(0)}G(x,y)\Delta u(y)dy$$ where $G$ is the Green's ...
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### Can complex analysis be used to solve Laplace's equation in three dimensions?

Can complex analysis be used to solve Laplace's equation in three dimensions? Or is it restricted to cases which have 2D symmetry? All of the examples I've seen are 2D, never 3D. It would seem that ...
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### Maximum Principle of Laplace equations

$u$ is the $C^2$ solution of $$\begin{cases}\Delta u = 0 &\text{in }\mathbb{R}^d\backslash B_R\\ u = 0 & \text{on } \partial B_R\end{cases}$$ So the problem asks to show that ...
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### How can we prove the harmonic mean is concave function?

For the function $f = \frac{n}{\sum_i 1/x_i}$ which is actually the harmonic mean of the values $x_1,x_2,\ldots,x_n$. How can we prove this is concave function? Because this is not a function of one ...
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### Is there any way to calculate harmonic or geometric mean having probability density function?

I have probability density of function of some data (it's triangular.) How can I calculate harmonic or geometric mean of the data? I know for calculating arithmetic mean of a variable like $K$, I have ...
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### Behavior of Poisson Integral Formula for Half Space

I am trying to solve the following: If $g$ is bounded and $g(x)=|x|$ for $x\in \mathbb{R}^{n-1}$ such that $|x|\leq 1$, then $Du$ is unbounded in a neighboorhood of zero. Remembering that: the ...
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### Proof for Mean Value Property using a specific limit

I am trying to prove the following: Suppose $u \in C^2(\Omega)$. For some $x \in \Omega$ we have that \begin{align} \Delta u(x) = \lim_{r \to 0} \frac{2n}{r^2} \left[ \frac{1}{w_n} \int_{\...
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### Show that $u(z)=\frac{1-|z|^2}{|1-z|^2}$ is harmonic in the open disc with center 0 and radius 1

Show that the function $u(z)=\frac{1-|z|^2}{|1-z|^2}$ is harmonic in $K(0,1)$: the open disc with center 0 and radius 1, and the determine the conjugate harmonic functions to $u$. I've been given the ...
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### Local barrier can be extended to a barrier

If $\Omega, \Omega_1$ are bounded, open, connected subsets of $\mathbb{R}^n$ with $\Omega_1\subset \subset \Omega$ (alternatively, $\overline{\Omega_1}\subset \Omega$), $\zeta\in \partial \Omega_1$ ...
### $\Delta u = 1$ and $u$ convex $\implies$ $u$ quadratic polynomial
Let $u \colon \mathbb{R}^{n} \to \mathbb{R} \in C^{4}(\mathbb{R}^{n})$ be a convex function such that $\Delta u = 1$. Show that $u$ is a quadratic polynomial. What I have done: Every harmonic ...
Consider a one-dimensional elliptic second order differential equation on an interval $[a, b]$. I am specifically interested in Sturm-Liouville problems where the principal symbol is the Laplacian and ...