# Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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### Physical Intuition for Mean Value Property of Heat Equation.

I am trying to think of an intuitive explanation for the mean value property for the heat equation: $$u(x,t) = \frac{1}{4r^2} \int_{E(x,t,r)} u(y,s) \frac{|y|^2}{|s|^2} dy ds$$ where $E(x,t,r)$ is the ...
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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 ...
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### Harmonic function such that $u =\partial_\nu u = 0$ on a surface must vanish.

I'm studying problems from some qualifiers and I came across a problem I couldn't solve: Let $\Omega\subseteq \mathbb{R}^3$ be a domain and suppose that $\Sigma\subseteq \Omega$ is a smooth (...
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### Holomorphicity of $\sin(\theta)\cos(\theta)+i\cos(\theta)^2$ imply that $\theta$ is constant

Suppose $\theta(x,y)$ be a smooth function on a connected domain, taking values in $(0,2\pi]$, and define $c=\cos(\theta),s=\sin(\theta).$ Suppose that the function $f(x,y)=sc+ic^2$ is holomorphic. ...
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### Fourier Transform of a harmonic polynomial

Let $x \in S^n$ and $P_d$ be a homogenous harmonic polynomial of degree $d$ in $n+1$ variables. I would like to find Fourier transform of the following function: $$\frac{1}{|x|^{n+d}}P_d(x).$$
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### Prove: $\sum\limits_{j=1}^{2^n}\frac{1}{j} \geq 1 + \frac{n}{2}$ for all positive integers $n$.

Given the following proof. Proof: by induction on n Base Case: $n = 1$ is trivial. Done Inductive Step: suppose for $n \geq 1$ we have $\sum\limits_{j=1}^{2^n}\frac{1}{j} \geq 1 + \frac{n}{2}$. We ...
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### Can I use the mean value property for a function that is harmonic on only the interior of a ball?

Let $f$ be a continuous function on the closure of $U$ where $U=\{ (x,y) \in \mathbb{R}^2 : x^2+y^2<1 \}$ and harmonic on $U$, If $$f(x,y)=x^2y^2$$ on $\partial U$ I want to find $f(0,0)$. Now I ...
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### Laplace's equation on the plane: how much boundary data must be specified to guarantee existence and uniqueness?

My question stemmed from a specific problem, so let's jump right in. I want to solve Laplace's equation in plane polars $(r, \theta)$ on the domain $(r,\theta) \in [1,\infty)\times[0,2\pi)$ subject to ...
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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 ...
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### harmonic function in a 2d disc.

So my professor gave as the following question: Given the function u which is continuous in $\bar{D}$ , where $D = \left \{ (x,y) \in \mathbb{R}^{2} : x^{2}+y^{2}<1 \right \}$, and harmonic in $D$. ...
I'm trying to solve exercise 2.9 of Axler's book "Harmonic function theory". The exercise says: Show that if $u$ is a pointwise limit of a sequence of harmonic functions on $\Omega$, then \$...