Mathematics Stack Exchange
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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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Harmonic functions theorem

I have to prove this theorem but monstrous computation appear when i try to compute the laplacian of $\hat{u}$. Does anyone know an easier way for the proof? Let be $\Omega$ an open subset in $\mathbb{R}^{n}$ and $u:\Omega \to…
Ant2198
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Weighted averages in harmonic functions.

Is it the case that for a harmonic function on a graph any value of the interior point is the weighted average of the boundary points? I know that for a harmonic function each point is the weighted average of its adjacent neighbors, but can you…
Pam
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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 Property. Since Cauchy-Schwarz only works for $p=2$,…
phantom_dino
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Determine amplitude and period of an added cosine

Graph $g(t)=2\cos(\frac{\pi}{3}t)$ (dashed curve) $f(t)=g(t) + Acos(\omega t)$ (drawn-through curve) Obviously $A = 1$. But what is the value of $\omega$?
QWERTYZ
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Proof that $\frac{u_x v_x - u_y v_y}{(u_x^2 + u_y^2)(v_x^2 + v_y^2)}$ is harmonic if $u$ and $v$ are

Can any one help me with this question? Let $u$ and $v$ be harmonic functions in a domain ($D$ is a simply connected space and open). Assume $(u_x^2 + u_y^2)(v_x^2 + v_y^2) \ne 0$. Define $$\psi = \frac{u_x v_x - u_y v_y}{(u_x^2 + u_y^2)(v_x^2 +…
Omer Ben
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Calculate the values of t for which the harmonic oscillator function $e^{-ft}y\sin ω t$ has a minimum...

The problem on my hw reads as $e^{-y}\sin t$ (from the hw he hands out) with just a negative sign on the exponent instead of $e^{-ft}y\sin ω t$ where $f$ and $ω$ are real constants; $f$ = friction & $ω$ = frequency. (from the online notes) Did he…
stack ex
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Harmonic function 2

Let $P$ be a harmonic function show that : $$\Delta P = 4\frac{\partial^2 P}{\partial z\partial \overline{z}}$$ I have no idea how to even start, please if someone can push me in the right way that would be great.
the_firehawk
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maximum principle for harmonic functions (equivalence)

If $f$ is harmonic on an open set then the maximum and minimum principle ($M$) applys to $f$. Are there non trivial known properties $A(f)$ s.t. following holds: Let $U\subset\mathbb R^n$ open and $f:U\rightarrow\mathbb R$ then $A(f)\wedge…
cousle
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Harmonic function and maximum modulus principle

Does Maximum modulus principle hold true for harmonic functions? I think we need open mapping theorem for this condition to hold true?
Avadutta
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suppose $u$ and $v$ are real harmonic function in a plane region $\omega$

Suppose $u$ and $v$ are real harmonic function in a plane region $\omega$. Under what conditions is $uv$ harmonic?
user395981
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Some questions about harmonic functions

-Assume $u$ is a harmonic function on $\mathbb{R}^3$, and assume u(x,y,z)=1+x when $x^2+y^2+z^2=1$. What is the value $u(0,0,0)$? I am in doubt between $1$ and $0$, could it be both? -Let $V(r)$ be a radial harmonic function in $\mathbb{R}^3$;…
Cippa Lippa
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Could a non-constant harmonic function be bounded or has extrema ? Could it exist in the physical world?

Harmonic function is a function which its Laplacian is equal zero: $$ {\displaystyle \Delta f=\sum _{i=1}^{n}{\frac {\partial ^{2}f}{\partial x_{i}^{2}}}} =0$$ Harmonic functions have the mean value property which states that the average value of…
Mohamed Mostafa
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Value of $f_x g_x+ f_y g_y + f_z g_z$ when $f$ and $g$ are harmonic

Let $f$ and $g$ be distinct real-valued harmonic functions, which not merely differ by a constant or are not merely multiples of each other. Also, assume that the first order partial derivatives of the two functions do not vanish identically. Is it…
vnd
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Question on real-valued harmonic function

Let $V\subset\mathbb{C}$ be a connected open set and $u$ a real-valued harmonic function on $V$. Suppose that the set $S=\{p\in V \mid \nabla u(p)=0\}$ has a limit point in $V$, then $u$ is constant. My solution: Let $q \in V$ be a limit point of…
Aleph-null
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Biharmonic operator; properties, identities

The biharmonic operator is $\nabla^4 \phi \equiv \nabla^2 (\nabla^2 \phi)$. Are there any identities for it? I need to find $\phi$ such that $~\\$ $\nabla^4 \phi = \frac{1}{3}\nabla^4 u^3 - u \nabla^4 u^2$, where we know $\nabla^2 u = 0$.…
user251041
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