Tagged Questions

For questions regarding harmonic functions.

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Why is a harmonic conjugate unique up to adding a constant?

If $v$ and $v_0$ are harmonic conjugates of $u$, then $u + iv$ and $u + iv_0$ are analytic functions. Then $i(v - v_0)$ is analytic, but how does this imply $v - v_0$ is a constant function?
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Laplace's equation in an infinite strip - bounded vs. unbounded solution

Let $D = \{\ ( x,y )\in \mathbb{R}^{2} \ | \ 0<y<1\ \}$. Let $A,B\in\mathbb{R}$. Consider the boundary value problem: $\Delta u = 0$ $u(x,0)=A\$ and $\ u(x,1)=B\$ for all $x \in \mathbb{R}$ ...
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Net flux zero equivalent to vanishing solution?

Consider \begin{align*} \Delta u(P) = 0, \quad P\in \Omega,\\ \frac{\partial u(P)}{\partial n_{p}} = f(P),\quad P\in \Gamma, \end{align*} where $\Omega$ is the infinite domain in the plane outside ...
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Let $u(x)$ be harmonic for $x \in \Bbb R^3$. Suppose that $\nabla u \in L^2 (\Bbb R^3)$. Prove that $u$ is a constant.

Let $u(x)$ be harmonic for $x \in \Bbb R^3$. Suppose that $\nabla u \in L^2 (\Bbb R^3)$. Prove that $u$ is a constant. I tried with Green's equation but it didn't work. Could anyone tell me which ...
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Prove $u$ is non constant on each disk $D(r)$ for all $0 <r<1$ and $\varphi (r)$ is strictly increasing in $[0,1]$.

Let $D(r)=\{(x,y):x^2+y^2\le r^2\}$, and let $u(x,y)$ be a non constant harmonic function in the unit disk $D(1)$. Let $\varphi(r)=\max_{(x,y)\in\partial D(r)}u(x,y)$ for $0\le r\le 1$. Prove ...
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Deriving the Equation for the Depth to Which an Object Falls

For a body of mass m kg, show that the depth to which the body would fall if attached to a rope, with a length of l meters. The depth is given by the model: d= \frac{2ml \pm l \cdot \sqrt{4m^2 + ...
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harmonic functions on the disk which agree on the real are identical?

The question is whether it is possible to find two distinct harmonic functions on the unit disk $\mathbb{D}=\{z: \ |z|<1 \}$ such that they agree on $\mathbb{D} \cap \mathbb{R}$. If yes, please ...
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Uniformly bounded family of harmonic functions

I am pretty sure other questions on this site can answer this problem, but I'm really interested in knowing if this particular solution is valid. Thanks. Question: Let $U$ consists of the set of real-...
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Describing the zero level set of a harmonic function

Let $p(k)$ be a complex polynomial of degree $n\in\mathbb{N}$. Let $A=\{k\in\mathbb{C}:\text{Re}\,p(k)=0\}$ The harmonic function $\text{Re}\,p(k)$ determines the behaviour of $A$. Fix $z\in A$. If ...
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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 $\mathbb{R^{2}}$?
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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. ...
"If $u$ is harmonic and bounded in the punctured disk $0<|z|< \rho$, then $u$ can be extended harmonically to the disk $|z|<\rho$ harmonically." This fact has been shown here. My Question is:...