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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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The unique existence of a harmonic function $u$ that agrees with a continuous function $h$ on the smooth boundary of a domain

I'm reading below statement at page 36 of these notes, i.e., Fact. Let $D$ be an open and bounded domain in $\mathbb{R}^n$ and $\partial D$ be its (smooth) boundary. Let $h \in \mathcal{C}(\partial D)$. Then there exists a unique function $u \in…
Akira
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Question about harmonic functions on the complement of a bounded set.

I'm trying to prove 2.74 Theorem of Folland's book, Introduction to PDE. It says: If $u$ is harmonic on the complement of a bounded set in $\mathbb{R}^n$, the following are equivalent: a) $u$ is harmonic at infinity. b) $u(x)\to0$ as $x\to\infty$ if…
ifthenelse
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show that $|x|^{-(n-2)}$ is harmonic function

let $f:\mathbb{R}^n \to \mathbb{R}$, and $f: \mathbf{x} \mapsto \Vert\mathbf{x}\Vert^{-(n-2)}$ show that $f$ is harmonic. I tried to take derivative by $x_1$ twice, and I got this result: $1/4 (-1 + 4 n^2) (x + x_{2...n})^{(-3/2 - n)}$ this of…
hash man
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Bounded harmonic function on slit upper half-plane

Let $$\Omega = \{z \in \mathbb{C} : \textrm{Im}(z) > 0\} \setminus \{iy : y \geq 1\}.$$ I need to find a bounded harmonic function $u \colon \Omega \to \mathbb{R}$ such that for each $x \in \mathbb{R}$ we have $u(x + iy) \to 0$ and $u(z) \to 1$ as…
Ethan Alwaise
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F. Riesz Theorem on harmonic and subharmonic functions

In the book "Uniform Algebras and Jensen Measures" by T.W. Gamelin, p.39, says: $---------------------$ By the F.Riesz Theorem, any subharmonic function $u$ in a neighborhood of a compact set $K$ in $C$ can be expressed in the form $$u(z)= v(z) +…
LDS
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On the partial derivatives of a harmonic function

Well, here is the thing. We know that the laplacian operator commutes with any partial derivative of a function, if the function is smooth. We also know that a harmonic function is infinitely differentiable, thus every partial derivative of a…
Lessa121
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Properties of harmonic functions

If $u:D \mapsto \mathbb{R}$ and $v:D \mapsto \mathbb{R}$ are harmonic functions, then also function $uv:D \mapsto \mathbb{R}$ is harmonic. Is the statement correct?
ELEC
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Harmonic Function With Step Function Boundary Data

Consider the Unit Disk. can we solve for a harmonic function in the unit disk such that: $\triangle u = 0 $ in D and $ u = f $ on $\partial D$ where $ f = 1$ for $|\theta| \leq \epsilon$ and $ |\theta - \pi| \leq \epsilon$ and $f=0$ everywhere…
Ali
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Examples for 2-dimensional real valued harmonic functions

Given: $$f: \mathbb{R}^2 \to \mathbb{R},\space \Delta{f}=0, \space\frac{\partial^2{f}}{\partial{x}^2_i} \neq 0, i \in\{1,2 \}.$$ Are there examples of such functions?
user76568
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square of a harmonic function bound

I need to solve this problem: Let $u$ be a harmonic function inside the open disk $K$ centered at the origin with radius $a$. We are also given that $\int_K u^2(x,y)dxdy=M<\infty.$ Show that $|u(x,y)|\leq…
arestes
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Harmonic function on upper-half space

Let $H=\{(x,y,z)\in\mathbb R^3\,|\,z\geq 0\}$, let $f:H\to\mathbb R$ be harmonic on the interior of $H$, and let $f$ satisfy the boundary condition $f(x,y,0) = a$ for some $a\in\mathbb R$. One easily verifies that $f(x,y,z) = a+bz$ satisfies all of…
joshphysics
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Harmonic functions which vanish on the real line

Let $u:\mathbb{C}\to \mathbb{R}$ be harmonic on the entire plane. Is there a nice classification, maybe a nice basis, for the set of such functions that satisfy $u\equiv 0$ on $\mathbb{R}$? Examples: $u(x,y)=0$ $u(x,y)=y$ $u(x,y)=\mathrm{Im}(…
Ed M
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Sequence of non-negative harmonic functions

Here is the text of the problem I'm attempting: Suppose that $(u_j)$ is a sequence of non-negative harmonic functions on $B_1(0)$ such that the sequence of numbers $u_j(0)$ converges. Show that there is a non-negative harmonic function u on…
Abrb
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Mean value property for 'almost' harmonic function

Let $B_r(x)$ be an open ball in $\mathbb R^n$ centered at $x$ with radius $r$. In the literature, it is known that if $u \in C^2(B_{2}(0))$ and $u$ is harmonic in the sense that $\Delta^2u = 0$ everywhere, then $u$ has the mean value property,…
QA Ngô
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Show that the Laplace equation is rotation invariant.

Let $u(x)$ be a harmonic function in $\mathbb{R^n}$ and $A$ an orthogonal matrix then I wish to show that $u(Ax)$ is also harmonic. I understand how to do this by showing that $$\triangle{u(Ax)}=0$$ But would like to show this using the Mean value…
OEB
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