Given a smooth $m$-dimensional manifold $\mathcal{M}$ embedded in $\Re^k$. Suppose we have a map $f : M \to \Re .$
Now, these are my questions:
Specific question:
i): Why does the $f$ that minimizes $\int_\mathcal{M}\| \nabla f(x)\|^2$ have to be an eigenfunction of $ \operatorname{div}\nabla(f)$? Let's assume that the minimization is done under the constraint $\| f \|_\mathcal{L^2(M)}=1$.
More Generic question:
ii) If the map was instead vector-valued as in $f : M \to \Re ^n$ and if the constraint was replaced by a condition that it satisfies a linear system, as in $f(x)y=b$, where y and b are given, then how does the minimizer relate w.r.t the eigenfunction solution in i)? Can the minimizer still be called an eigenfunction?. How does the minimizer relate to the eigenfunction in i) if the constraint was another surface instead of a plane-as in say a hypersphere?
Please include as much detail as possible on your explanations. Thanks.
i) Because the Laplace-Beltrami operator $\Delta$ is a self-adjoint operator on $L^2(M)$ as soon $M$ satisfies your assumptions, its lowest eigenvalue is 0 (the constants being the null space of $\Delta$). The second lowest eigenvalue is then, according to the Courant-Fischer Theorem, the minimiser of the Rayleigh quotient $\frac{\|\nabla u\|^2}{\|u\|^2}$ under the further condition that $u$ is orthogonal to the constants. By homogeneity of the Rayleigh quotient (or equivalently, by linearity of $\Delta$), one can just minimise the functional $\|\nabla u\|^2$ over all normalised functions, since the orthogonality conditions is then automatically satisfied.
ii) I hope the above answer also explains why your first idea is hopeless. No matter whether $\Delta$ acts on scalar or vector-valued functions, what you need in order to apply Courant-Fischer is a condition that is equivalent to orthogonality to the constants (with respect to the inner product of $L^2(M)$). This cannot be implemented by a linear system that has to be satisfied at finitely many points. In fact, you certainly need to minimise the functional over a subspace of codimension 1 (or, using linearity, over a compact subset of a subspace of codimension 1).