# Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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### Understanding construction of fundamental solution to laplace's equation

We take a look at Laplace's equation $\Delta u=0, u:\mathbb{R}^n\rightarrow\mathbb{R}$ and want to look for explicit solutions, firstly, since Laplace's equation is invariant under rotations, for ...
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### Question on Theorem 2.2.13 in Partial Differential Equations by Evans

I'm working through the section on Laplace's Equation in Partial Differential Equations by Evans. I'm having trouble following a step in Evans's proof of the symmetry of Green's function (Theorem 2.2....
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### How to solve the two-dimensional Laplace equation in this unbounded and multiply-connected domain?

I have the Laplace equation $$\nabla^2 \phi(x,y) = 0$$ where $\phi$ is a function defined on the domain $\mathbb{R^2}\setminus(C_1 \cup C_2)$. $C_1$ and $C_2$ are two circles of radius $r = 0.5$ and ...
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### Solve $\nabla^2 u(x,y) = 1-y$ + B.C.

I am trying to solve: $$\nabla^2 u(x,y) = \partial_{xx} u(x,y) + \partial_{yy} u(x,y) = 1-y$$ over the domain $[-1,1]\times [0,1]$, with the B.C.: $$u(x,0) = 0 = u(\pm 1, y).$$ Is there any closed-...
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### Question about Theorem 1 in Chapter 2 of Evans's Partial Differential Equations

I am working through Partial Differential Equations by Evans and I am struggling to understand a step of his proof of theorem 1 in chapter 2. He presents the following argument: The function $\phi$ ...
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### Laplace equation can implies some result

Prove that $$\Delta u(x)=\Delta u(x_1,\ldots,x_n)=0$$ also implies that $$\Delta (|x|^{2-n}u(x/|x|^2))=0$$ for $x/|x|^2$ in the domain of definition of $u$. I have no idea how to start with this ...
1 vote
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### Solving an eliptical PDE in a rectangle

I was trying to solve this and got stuck right before the final solution, so that's the problem:  \begin{matrix} &u_{xx}+u_{yy}=0 & x\in (0,2), y\in (0,1) \\ &u(x,0)=0 & x\in[0,2]\...