The laplace operator is defined for the sphere as : \begin{equation} \frac{\partial^2 f}{\partial r^2} +\frac{2}{r} \frac{\partial f}{\partial r} +\frac{1}{r^2\sin\theta} \frac{\partial }{\partial \theta}\left( \sin\theta\frac{\partial f}{\partial \theta} \right) +\frac{1}{r^2\sin^2\theta}\frac{\partial^2 f}{\partial \varphi^2} \end{equation} Is it possible to have a definition for an half sphere space ?
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Can't you just use the same formula as the sphere case? – chan kifung Jul 12 '17 at 20:30
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No, the base founded on such operator would be on a sphere, i wonder if it's possible to find an half sphere basis. For example on laplace equation on sphere : $\triangle f = 0$ the solutions would be spherical harmonics but on half sphere these solutions would not be good – Boulgour Jul 12 '17 at 20:36
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What do you mean by a "basis", do you mean "coordinates"? $r,\theta,\phi$ can be seen as coordinates on a half sphere, just you need to restrict the domain of definition of $\theta$ to $[0, \frac{\pi}{2}]$. – chan kifung Jul 12 '17 at 20:42
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what do you mean by saying that the solutions to $\Delta f=0$ is not good? They are still harmonic functions with respect to the standard metric, just the harmonic functions on the non-compact half sphere may not be constant. – chan kifung Jul 12 '17 at 20:47
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@chankifung It's not that it's not good it's just that the basis defined by the spherical harmonics are defined for a sphere, what i would come up with is a kind of hemispherical basis, a basis defined for the hemisphere. Let's take the solution of the sphere for the laplacian equation for interior problem $\triangle f= 0$: $f(r,\theta,\phi) = \sum_{m=0}^{\infty}r^m\sum_{n=-m}^{m}A_{mn}Y_{mn}(\theta,\phi)$, if you know the potential at for a sphere of defined radius, it's possible by projection to define the A_{mn} then extrapolate. – Boulgour Jul 13 '17 at 01:14
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I would like to do the same with a half sphere, is that clearer? – Boulgour Jul 13 '17 at 01:14