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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.

2116 questions
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Is the harmonic function constant?

Suppose $f$ is harmonic on $\mathbb{R^{2}}$ and constant on a neighbourhood in $\mathbb{R^{2}}$. Is $f$ constant on $\mathbb{R^{2}}$?
BasicUser
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Methods for finding harmonic conjugate function

What are the methods for finding harmonic conjugate function? There is the Cauchy-Riemann equations but are there any other methods?
user223740
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To Find the Transfer Function Z(s)/X(s) for the system....

Please, help me to answer the next problem: Objective: To find the Transfer Function $z(s)/x(s)$ for the system, using the next equations: "$a$", "$b$", "$c$" y "$k$" are constants $x(t) = a y(t) + b y'(t)$ $w(t) = k y(t)$ $w(t) = c z(t) + g…
Arturo Ortiz
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A functional equation for harmonic functions

Does there exist a non zero function $u\in C(\mathbb{C})$, harmonic in $\mathbb{C}\setminus\mathbb{T}$ that satisfies the following equation: $$u(z)+u(-z-2)=0\:\:\forall z\in\overline{\mathbb{D}}$$ Where $\mathbb{D}=\{z: |z|<1\}$ and…
BigM
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Limit of natural log

Prove that $\displaystyle \lim_{n \to \infty} \ln x = \infty$ using the fact that the harmonic series diverges Of course, this is obvious graphically, but I have to prove it formally. I based my thinking on this comment: But I don't understand…
kiwifruit
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Harmonic function

Let $B(0;1)=\{x \in \mathbb{R}^N;|x|≤1\}$, the ball of $\mathbb{R}^N$ equipped with the euclidian scalar product $$x⋅y=x_1y_1+...+x_Ny_N,\ \ \ x=(x_1,...,x_N),\ \ \ y=(y_1,...,y_N)\ \ \ |x|=\sqrt{x \cdot x}$$ Let $\alpha$ and $\beta$ two real…
Student
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Choose parameters to make a harmonic function

Let $B(0;1)=\{x\in \mathbb{R}^N ;|x|\leq 1\}$, the ball of $\mathbb{R}^N$ equipped with the euclidian scalar product $$x \cdot y=x_1y_1+...+x_Ny_N,\ \ \ x=(x_1,...,x_N),\ \ y=(y_1,...,y_N)\ \ \ |x|=\sqrt{x\cdot x}.$$ Let $\alpha\mbox{ , } \beta…
Student
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Harmonic functions properties

Let $S (0,0) \subset \mathbb{R}^2 $ be the circular disc with radius $r=1$ and center $(0,0)$. Furthermore let $u_1(x,y)$ and $u_2(x,y)$ be solutions of: $\Delta u_n =0 \quad (x,y) \in S(0,0)$ $u_n =f_n \quad (x,y) \in \partial S(0,0)$ Where…
Andrew142
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Sub- and superharmonic functions

Let $\Omega \subset \mathbb{R}^n$ and $f\in C^2(\Omega)$ such that $\Delta f(x)\leq 0$ for some $x\in\Omega$. Thus, can we derive: $f(x)\geq \frac{1}{|B_r(x)|}\displaystyle\int_{B_r(x)}f(y)\,dy $ where $B_r(x)\Subset \Omega$? Or is there really…
HelloEveryone
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$\ln |f|$ is harmonic where $f$ is holomorphic

In a book, there is a proposition states that : Let $U$ be an open set of $\mathbb{C}$ and $f$ be a holomorphic function on $U$ does not take value $0$. Then $\ln |f|$ is harmonic. In the proof, it's written $\frac{\partial}{\partial z} \ln|f| =…
ZENG
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Laplace's Equation looking for vertical displacement of a membrane

Consider a two dimensional flexible membrane whose equilibrium position occupies a region in the horizontal x y-plane. Suppose this membrane vibrates up and down with u(x, y,t) denoting the vertical displacement of the point (x, y) of the membrane…
user927154
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$u$ harmonic and $\lim_{|z| \to \infty} u(z) = 0$ imply that $u \equiv 0$ in $\mathbb{C}$.

If $u$ is harmonic and bounded in $\mathbb{C}$, then I've shown that $u$ is constant. I guess it can be helpful to show that $u \equiv 0$ if $u$ is harmonic and $\lim_{|z| \to \infty} u(z) = 0$, but how? I guess $u$ is not neccesary bounded in…
joseabp91
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show that $u$ is harmonic with a condition

Let $D$ be a domain in $\mathbb{C}$ and let $u : D \to \mathbb{R}$ be a continuous function. I suppose that for each $a \in D$ there exists $r_a > 0$ with $\overline{D}(a , r_a) \subset D$ and such that $$ u(a) = \frac1{\pi r^2} \int…
joseabp91
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show that $u(r e^{i \theta}) = \theta \log r$ is harmonic in $\mathbb{C} \setminus (- \infty , 0]$.

My attempt to show that $u(r e^{i \theta}) = \theta \log r$ is harmonic is the next: if $z = r e^{i \theta}$ we have $\log z = \log r + i \theta$, then ${(\log z)}^2 = {(\log r)}^2 + i 2 \theta \log r - {\theta}^2 = {(\log r)}^2 + i 2 u(z) -…
joseabp91
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Does there exist a 2D harmonic function which never goes to ∞ at any point, goes to 0 as r goes to ∞, and isn't f=0?

I need a 2D harmonic function which is always real, never diverges to $\infty$ anywhere, goes to 0 as one goes infinitely far away from the origin, and isn't…
Laff70
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