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.

2116 questions
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What are the parameters of the following sine wave

I hope you can help me out because this problem has been bothering me all day long. Graph of sine function The tasks: A) The graph belongs to the function $f(x)=a\sin(bx+c)$. Determine the parameters a, b and c according to the graph. B) Determine…
QWERTYZ
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Harmonic functions and extrema property for spherical components

If a vector function $\bf A$ is harmonic (that is $\nabla^2 {\bf A} = 0$) on a region $V$ then in Cartesian coordinates each component of $\bf A$ is also harmonic on $V$. That is $\nabla^2 A_i = 0$ where $i=x,y$ or $z$. It then follows from the…
D_J_S
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Show that the equation defines a harmonic conjugate

Suppose $u$ is a twice continuously differentiable real-valued harmonic function on a disk $D(z_0;r)$ centered at $z_0 = x_0 +iy_0$. For $(x_1, y_1) \in D(z_0;r)$, show that the equation \begin{align*} v(x_1, y_1) = c + \int_{y_0}^{y_1}…
mXdX
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What's the term involving $fw$ (or $fv$) in Laplace's eq?

I have some confusion because I'm given the form $$\nabla^2 u = 0$$ of Laplace's eq. However, I'm viewing a reference that seems to have some sort of integral involving terms $fw$ (a product of $f$ and $w$, some refs seem to write it as $fv$) on the…
mavavilj
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Solving Laplace's Equation on unit sphere

I have reduced a calculus of variations problem into solving laplaces equation in 3D in the unit sphere with boundary conditions $u=1$ on the boundary of the sphere and $\int\int\int u dV = 4\pi$ when integrated on the unit sphere. Is there a simple…
Rich N.
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Solving Laplace's equation within an infinite domain

I try to solve Laplace's equation $\Delta u=0$ in the half plane: $${d^2u\over dx^2}+{d^2u\over dy^2}\:=\:0 \quad\text{where } -\infty0 $$ and the boundary condition $$\left.u\right|_{y=0}\:=\:\frac{x^2-1}{(x^2+1)^2} = \frac…
Ntr
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Limit of Harmonic function

Let V be a harmonic and bounded function on $\mathbb{R}^2 \setminus \text{B}_R,\text{B}_R $ -- a ball with radius $R$. Proof that: 1) $\exists \lim\limits_{x \to \infty} V(x) $ 2) $\nabla V(x) = \mathcal{O} \left( \frac{1}{|x|^2} \right), x \to…
Kamil
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harmonic function

Let $f$ be a real function with $\Delta f=0$ on an open ball $B_{2n}(y)\subset\mathbb{R}^N$. How would I show $$\int\limits_{B_n(y)}|Df|^2(z)dz\leq Cn\int\limits_{\partial B_n(y)}|Df|^2(z)d\sigma(z)$$ for some constant C, where $\sigma$ is surface…
Katie
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Show that $Q$ is harmonic

Let $P : D\subset \mathbb{R}^2 \to \mathbb{R}$ be a harmonic function show in both cases that $Q$ is harmonic first case $Q = x\frac{\partial P}{\partial x}+y\frac{\partial P}{\partial y}$ $$ \Delta Q = \frac{\partial^2 Q}{\partial…
the_firehawk
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harmonic function on upper half sphere

Let $B^+:=\{x=(x',x_d) \in \mathbb{R}^{n-1} \times \mathbb{R} :|x| < 1, x_d>0 \}$ the upper half sphere. Let $u \in C(\bar{B^+}) \cap C^2(B^+) $ harmonic on $B^+$ and $u(x)=0$ for $x=(x',0)$. Let $v(x',x_d) := u(x',x_d)$ for $x_d \geq 0$ and…
Hamilcar
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tangent plane to graph of a harmonic function

Suppose $u \in C^2(\omega)$ harmonic, with $\omega$ open and connected. Let $\{(x,y,v(x,y)), (x,y) \in \omega\} $ be the tangent plane in $(x_0,y_0)\in \omega$ to the graph of $u$, $G(u)=\{(x,y,u(x,y)), (x,y) \in \omega\} $. One has to prove that…
Romanda de Gore
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$|Du|^2$ is subharmonic if $u$ is harmonic.

In Evan's textbook "Partial Differential Equation", question 5 in section 2.5 says "$|Du|^2$ is subharmonic if $u$ is harmonic.". This can be easily proven, but do we really need the derivative $D$? I don't see any point of taking the derivative of…
Pooya
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I need to prove that this function is harmonic! [Solved]

I need to prove that $u:\mathbb{R}\times(-\frac{\pi}{2},\frac{\pi}{2})\rightarrow\mathbb{R}$ $$u(x,y)=\sum_{n \ \text{ is odd}}\cos(ny)e^{n(x-n)}$$ is harmonic. I have no idea which theorem or result to use.
José Carlos
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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 value property, but this what only introduced later…
M.G
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Harmonic Conjugate (Multiplication, power, addition/subtraction)

The Question Suppose that $v$ is a harmonic conjugate for $u$ on a domain $D$. Prove that $u(x,y)^3 - 3u(x,y)v(x,y)^2$ is harmonic. I'm trying to prove that this function is also harmonic when $v$ is harmonic conjugate for $u$. I know that when…
Cathy Kim
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