# Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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### Query about Laplace Eqution and Harmonic functions

In the text below, I am unable to understand how Eq 2.5 represents the Laplace Equation because I don't see any partial derivatives etc. Does Eq 2.5 somehow reduce to $\nabla^2 u=0$? The snippet is ...
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### Expansion for Harmonic Functions

Take $u\in W_\delta^{2,p}$(weighted Sobolev space), $\delta\notin \mathbb{Z}.$ If we have $$\Delta u=0,\quad |x|>R,$$ where $R$ is a constant. It's said that "the classical expansion for ...
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### Level curves of harmonic functions are analytic curves?

In the paper, of Flatto, Newman, Shapiro about level curves of harmonic functions, namely curves $\Gamma$ for which there exists a harmonic function $u(x,y)$ vanishing on $\Gamma$ but not identically,...
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### Green's Function Computation

I want to calculate Green's Function to solve $\triangle u = f(x,\ y)$, using Laplace Transforms. My plan was to tailor boundary conditions to the problem which simplify the computation. Because the ...
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### bounding the rate of the harmonic function can decay

Let $u$ be the harmonic function on $\Bbb{R}^n$, we can build the following estimate using the inverse Poincare inequality: $$\int_{B_{2r}}u^2 \ge (1+c(n)) \int_{B_r}u^2 \tag{*}$$ Where $c(n)$ is a ...
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### Constructing meromorphic differential over compact Riemann surface

On a compact Riemann surface, one can construct meromorphic differential having only simple poles by using dipole Green function by means of Perron method. (the construction is here and having such ...
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### Necessary and sufficient conditions for existence of solutions to $\Delta \phi = f$ on torus
Let $\mathcal{T}$ be a torus with Riemannian metric. Consider the sourced Laplace equation on $\mathcal{T}$: \begin{align} \tag{1} \Delta \phi = f. \end{align} I'd like to know necessary and ...