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 $\Delta f\left(x_{1},x_{2},x_{3}\right)=-\mu_{3}f\left(x_{1},x_{2},x_{3}\right)$ in the region enclosed by $S$ with $f$ vanishes on $S$? Thanks.

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.