Question

I have a question about eigenvalue problem on the geodesic ball in $n$-dimensional sphere $\mathbb{S}^n\subset\mathbb{R}^{n+1}$.

Consider the eigenvalue problem in the geodesic ball $\Omega=\{x_{n+1}\geq c\}$ where $c\geq 0$:

$\Delta u+nu=0$ in $\Omega$

$u=0$ on $\partial\Omega$.

For the upper hemisphere, i.e. when $c=0$ and $\Omega=\{x_{n+1}\geq 0\}$, the first eigenfunction is given by $u=x_{n+1}$. My question is: for $c>0$, can we find the first eigenfunction explicitly? Can we write down the expression of the first eigenfunction, say, in terms of the coordinate functions? I was told it is related to Legendre function. But I am still not sure how to write down the expression.

Answer 1

You have the following estimate (see (3.10) of Stability of minimal surfaces and eigenvalues of the Laplacian, Barbosa & do Carmo).

Let $D$ a simply connected domain of $S^2$,

if $\vert D\vert \leq 2\pi$ then $\lambda_1\geq \frac{4\pi}{\vert D\vert}$,

if $2\pi \leq \vert D\vert\leq 4\pi$ then $\lambda_1\leq \frac{2(4\pi-\vert D\vert)}{\vert D\vert}$.

You will find other estimate in the Book of Chavel: Isoperimetric inequalities.

Finally you can compute explicitly the solution since every thing is radial, but it depends of Bessel function if i remember well. Anyway there are solutions of an explicit second order o.d.e., which could be enough for numeric estimate.