Given an algebraic surface $S$ defined by an algebraic equation such as $x^{4}+2y^{4}+3z^{4}=1$, how would one find the third smallest eigenvalue $\mu_{3}$ for the differential equation $\Delta f\left(x,y,z\right)=-\mu_{3}f\left(x,y,z\right)$ in the region enclosed by $S$ with $f$ vanishes on $S$? Thanks.
4
-
$\begingroup$ Isn't this is a simple Laplacian eigenvalues problem? $\endgroup$– user21257Commented Mar 8, 2012 at 19:05
-
$\begingroup$ Find in what sense? Numerically? $\endgroup$– Igor RivinCommented Mar 8, 2012 at 22:31
-
$\begingroup$ @Igor. It would be both in symbolic sense and in numerical sense, but with symbolic sense preferred. $\endgroup$– user21990Commented Mar 9, 2012 at 16:39
-
$\begingroup$ Could you explain how to get the explicit eigenfunctions exactly? Thanks. $\endgroup$– user21990Commented Mar 9, 2012 at 16:40
Add a comment
|