Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

The solutions of the Laplace equation $\Delta f =0$ on a domain $D\subset \mathbb{R}^n$ are known as harmonic functions.

Harmonic functions appear most naturally in complex analysis and the Laplace equation is the most important PDE to study.

The Cauchy-Riemann equation together with the conjugated Cauchy-Riemann equation shows that the sum of an analytic function and an anti-analytic function is harmonic and in fact every complex harmonic function can be written as such. In particular the real/imaginary part of an analytic function is harmonic.

In any dimension, harmonic functions satisfy the following properties

  • Mean value property,

  • Maximum principle,

  • Harnack inequality,

  • Liouville's theorem.

Harmonic functions satisfy the regularity theorem for harmonic functions, which states that harmonic functions are infinitely differentiable (follows from Laplace's equation).

Please use instead the tag Laplacian if your question concern the Laplacian as an operator.

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Kato inequality

For any real-valued smooth function $u$, we have the Kato inequality $|\nabla|\nabla u||^2\leq(\operatorname{trace}(\operatorname{Hess}(u)))^2$, which holds when $|\nabla u|\neq0$. If moreover $u$ is harmonic (in an open set in $\mathbb{R}^n$), how…
Miranda
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Harmonic in punctured ball and bounded implies harmonic in ball.

If $u$ is harmonic and bounded in $B_1(0)\setminus\{0\}$, then can we say that $u$ is harmonic in $B_1(0)$? I believe the answer is yes and I think the way to show it is by the Mean Value Property... but there is a problem. For $|x|<1/2$ and…
Adam Martens
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Visualization of subharmonic functions

I have always visualized subharmonic functions as Ahlfors' Complex Analysis thaught me to do: in one dimension lines are harmonic functions and "convex" functions are subharmonic. I actually just tried to generalized this to multidimensional domains…
user67133
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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 $ F = \nabla f$ with zero divergence: $ \nabla F…
Asier Calbet
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A version of Casorati-Weierstrass for harmonic functions?

Suppose that $f:B(0,1)\setminus\left\{0\right\}\subset \mathbb{R}^n \to \mathbb{R}$ is a harmonic function. Clearly, the property that $\overline{f(B(0,\epsilon))}=\mathbb{R}$ for all $\epsilon>0$ is equivalent to $\limsup_{x \to 0 } f(x) = +\infty$…
5th decile
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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 $E$ and takes boundary value $F(x,\cdot)$.…
user118073
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Show that $H(x) := |x|^{-1} u(x/|x|^2) $ is harmonic if $u$ is harmonic

I have a "simple" question but I'm not able to solve it. Suppose that $u$ is a harmonic function on $\mathbb{R}^3$. Prove that the function $H(x) := |x|^{-1} u(x/|x|^2) $ is harmonic on $\mathbb{R}^3 \setminus\{0\}$ I tried brute force evaluating…
Gabrielek
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Showing that Poisson kernel for the unit disc is harmonic.

Let $r \in [0,1)$ and $\theta \in [-\pi,\pi]$ and define, $$P_r(\theta) = \frac{1-r^2}{2\pi(1+r^2-2r\cos(\theta))} = \frac{1}{2\pi}\sum_{n=-\infty}^{\infty}r^{|n|}e^{in\theta}.$$ Then I want to show that $\Delta P_r(\theta) = 0.$ Should I just use…
Student
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Show that the following function is harmonic [Solved]

I am trying to show that the function: $$u(x)=|x|^{(2-n)}$$ is harmonic where $x$ is a vector in $\mathbb{R}^n\setminus\{0\}$ Here is what I tried: $\displaystyle u(x)=|x|^{(2-n)}$ $\displaystyle $$\frac{\partial u(x)}{\partial x_{i}}=x_{i}(2-n)…
Charlie
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Constructing a specific harmonic function.

Is it possible to construct a $\bf{harmonic}$ function $u:\mathbb{R}^N\to \mathbb{R}$ satisfying: I - There exist a sequence $x_n\in \mathbb{R}^N$ such that $|x_n|\to\infty$ and $u(x_n)\to\infty$, II - There exist a sequence $y_n\in\mathbb{R}^N$…
Tomás
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Two harmonic functions with positive ratio are multiple of each-other?

Suppose $f,g,h:\mathbb{R}^n \to \mathbb{R}$ are functions so that $f$ and $g$ are harmonic and not identically zero, $f=g\cdot h$ and $h\geq 0$. Is $h$ a constant function? EDIT: Someone voted to close presumably because of a lack of context. The…
5th decile
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Harmonic functions which are constant on boundary

I know that a harmonic function $f\in C^2(\Omega)\cap C(\bar\Omega)$ which is constant on boundary is constant, if $\Omega\subset\mathbb{R}^N$ is a bounded and connected domain. Also, if $\Omega$ is not bounded, this property does no longer hold…
GlassFlake
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property of harmonic functions

If a real-valued function $u$ is harmonic on a ball $B_{2r}(x)$ in $\mathbb{R}^n$, how would one show that $$\sup_{B_r(x)}u^2\leq\frac{2^n}{|B_{2r}(x)|}\int_{B_{2r}(x)}u^2(y) dy$$
Audrey
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Logarithm of absolute value of a harmonic function subharmonic?

If $f(x)$ is harmonic, then is $\log|f(x)|$ a subharmonic function? I know if $f(x)$ is analytic, this is true, but for harmonic $f(x)$, is it still true? Thanks!
fengpeng wang
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Two question on harmonic function

In a question paper I got the following two questions. $u(z)=u(x,y)$ be a harmonic function in $\mathbb{C}$ satisfying $u(z)\le a|\ln|z||+b$ for some positive constants $a,b$ and for all complex numbers. Show that $u$ is constant. If for all…
Myshkin
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