# Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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### Poisson formula on a disc is continuous on the closure

It is written that the function $u$ is harmonic in $\mathbb{D}$ and continuous in $\overline{\mathbb{D}}$ and $u(e^{it})=\phi(t).$ I am unable to prove the continuity. z The continuity in the inside ...
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### Can a decreasing sequence of subharmonic functions converge to a discontinuous function?

If $u_n:\Bbb C\to \Bbb R$ are subharmonic, bounded, continuous, and $u_{n+1}\le u_n$ for all $n$, can their limit be discontinuous? By boundedness, it is easy to show that the limit function $u$ will ...
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### $Log(z^2)$ analytic for all of the complex plane except origin.

The question was show $\ln{x^2 + y^2}$ is harmonic in two ways. It was very easy to show by LaPlace's equation, but next I have to show it by showing it is the real part of an analytic function. I am ...
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### Estimates on Hessian of Solution to Poisson Equation

Let $\Phi:\mathbb{R}^d\to \mathbb{R}$ be the fundamental solution to the Laplace equation, i.e the unique function $\phi$ such that $\Delta \phi = \delta_0$ in the sense of distributions. The solution ...
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### Integral of positive part of harmonic function over circles goes to infinity

I have been thinking about this problem off and on for a couple days now to no avail. If in answering this question you could address the things I've tried so far as "on the right track", "not really ...
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### Geometric intuition for harmonic conjugate functions

It is known that given a harmonic function $u$ of class $C ^ {2}$ defined in a simply connected subset of $\mathbb{C}$ , we can find a function $v$ also harmonic, such that $f = u + iv$, is a ...
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Solve the boundary-value problem $∆u = 0$ (by this we mean $u_{xx} + u_{yy} = 0$) in the rectangle $0 < x < π$, $0 < y < 1$, with the boundary conditions $u(0,y) = 0$, $u(π,y) = g(y)$, $u(... 0answers 12 views ### Max vs. min bound for nonnegative harmonic function Problem: Let$\Omega$be an open, bounded, simply connected subset of$\mathbb{C}$and let$u \colon \Omega \to \mathbb{R}$be a nonnegative harmonic function. Show that for each compact subset$K \...
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I was reading the proof of the Rado–Kneser–Choquet Theorem. The statement is there in the image (taken from the book "Harmonic Mapping in the Plane, Duren page-$30$": In the proof, he shows that ...
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### Green's function for $\Omega=\{ (x_1,x_2,x_3)\in \mathbb{R}^3:x_2,x_3>0 \}$

Compute the Green's function for the Laplacian, for the region $$\Omega=\{ (x_1,x_2,x_3)\in \mathbb{R}^3:x_2,x_3>0 \}.$$ My approach is to use a reflection argument similar to the one used for ...
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I have a question regarding the uniqueness of the potential flow past a cylinder. Consider a two dimensional uniform potential flow in $x_1$-direction past the cylinder $B_R = \{ x = (x_1, x_2) \in \... 0answers 21 views ### Relation between Warping Function and Solution of Poisson's Equation In the solution of torsion problem for non-circular cross-sections, the warping function is defined. Without going into the details, the torsion constant is defined as follows:$J = \int_A \left[ \... 0answers 27 views ### Solve this PDE using Fourier Transform Question: Solve the following system using Fourier Transform: \begin{alignat}{2} & \nabla^2 f = 0 & z<0 \\ & \frac{\partial f}{\partial z} = \frac{\partial g}{\partial x} \quad \frac{\... 0answers 170 views ### If v is harmonic conjugate of u, then the harmonic conjugate of3u^2 − v^3$is harmonic conjugate of Attempt: I tried to use cr-equations,but integration becomes clumsy. Also, I tried to form another holomorphic function which have real part equal to$3u^2 − v^3$using any other holomorphic functions,... 1answer 106 views ### Does having harmonic functions$u(x,y)$and$v(x,y)$guarantee having an analytic function? Verify that each given function$u$is harmonic (in the region where it is defined) and then find a harmonic conjugate of$u$. (a)$u=y$I was able to verify that$u$is harmonic pretty easily. But ... 1answer 260 views ### Finding the harmonic conjugate of a function The problem: Find the harmonic conjugate of$G(x,y)= 2v^2(x,y)-2u^2(x,y)$My attempt to solving it I know that "If two given functions u and v are harmonic in a domain D and their first-order ... 1answer 58 views ### Find a and b values so a given function is harmonic The Problem: Let$f(z)=f(x+iy)=u(x,y)+iv(x,y)$be an entire function. If$g(z)=au^2(x,y) - bv^2(x,y)$find values for a and b so$g(z)$is a harmonic function. My attempt to find a solution: Well ... 0answers 29 views ### 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$(x_1, y_1) \in D(z_0;r), show that the equation \begin{align*... 0answers 55 views ### An estimate for the conjugate Poisson Integral on the unit disk. Letf \in L^1(\Bbb{T})$and$Q_r(\theta)=\frac{2r\sin{\theta}}{1-2r \cos{\theta}+r^2},r \in[0,1]$the conjugate Poisson kernels and$P_r(\theta)=\frac{1-r^2}{1-2r \cos{\theta}+r^2},r \in[0,1]$and$\...
Let $G$ be a discrete group. Consider $G^{\mathbb{N}}$, the space of all sequences in $G$ equipped with product $\sigma$-algebra. Let $m:G^{\mathbb{N}} \to G^{\mathbb{N}}$ be the multiplication map ...