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In a class, I'll teach the Liouville theorem for harmonic functions with finite Dirichlet integral. What kind of illustrations can I use to elucidate the meaning and proof of the theorem?

Note that a proof of this result is for example on MathOverflow: Prove Liouville theorem without using mean value property

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    $\begingroup$ does this help? mathoverflow.net/q/100750/11260 $\endgroup$
    – Carlo Beenakker
    Commented Jul 9, 2022 at 11:16
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    $\begingroup$ Another point of view is to use spherical harmonics; $u(r,\theta)= \sum_{k=0}^\infty a_k(r) \psi_k(\theta)$ and then you can solve for $a_k$ to get $$a_k(r) =C_k r^{\beta_k^+}+ D_k r^{\beta_k^-}$$ and then try and show $C_k=D_k=0$ (here the equation for $a_k$ is Euler and so $\beta_k^+,\beta_k^-$ are the associated roots and $ \psi_k(\theta)$ are the eigenfunctions of the Laplace-Beltrami operator on $S^{N-1}$.) $\endgroup$
    – Math604
    Commented Jul 10, 2022 at 4:17

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