Recall the stability operator $L=\Delta_\Sigma+Ric(N,N)+|A|^2$ for a minimal hypersurface $\Sigma^n\subset M^{n+1}$. Stability of $\Sigma$ is equivalent to non-negativity of the Rayleigh quotient $$\lambda_1:=\inf\{\frac{-\int_\Omega \eta L\eta}{\int_\Omega \eta^2}:\eta \in C^\infty_0(\Omega)\},$$ for every bounded domain $\Omega\subset \Sigma$. Lemma 1.34 states that we can set up this variational problem in a weak sense. Namely, for $$I:=\inf\{\frac{\int_\Omega |\nabla_\Sigma \eta|^2-Ric(N,N)\eta^2-|A|^2\eta^2}{\int_\Omega\eta^2}:\eta\in W^{1,2}_0(\Omega)\},$$ we have $\lambda_1=I$. Furthermore, achieving a $W^{1,2}_0$ minimizer $u$ gives a smooth solution to the Dirichlet eigenvalue problem $Lu=-\lambda_1 u$.
As standard practice for these sort of elliptic problems, one takes a $W^{1,2}_0$ sequence $\{\eta_j\}$ which goes to $I$, then use Rellich compactness to extract a subsequence which $L^2$ converges to some $\eta$. My question is essentially one on the exponents involved at this step. Sobolev embedding + Rellich compactness tell us that that the compact embedding into $W^{1,2}_0\subset\subset L^2$ is possible for every $n\geq 3$, since then the critical exponent is always greater than $2$,
$$\frac{2n}{n-2}>2.$$
However, it seems that this step fails for $n=2$. Since this dimension dependence wasn't explicitly mentioned in the book, I wanted to confirm that
- this weak setup indeed fails for $n=2$.
Furthermore, this result, combined with Harnack, is used to show that the first eigenfunction cannot change sign. Therefore, I would be interested to know if
- if there exists some smooth function $Lu=-\lambda_1 u$ with $u|_{\partial \Omega}\equiv 0$ on some domain $\Omega\subset \mathbb{R}^2$ which does change sign.
There is no dimension restriction. The question comes down to the statement that of the compact inclusion $W^{1,p}\subset\subset L^p$ without any dimension restrictions. The $p=n$ case comes down to $L^p$ inclusions on finite measure spaces + Rellich, as in here.