# Yau's conjecture on nodal sets for manifolds with boundary

I've just read a review paper about Yau's conjecture on nodal sets of the eigenfunctions for the Laplace operator on manifolds. Briefly, if $$\phi_\lambda$$, $$\lambda$$ are an eigenpair for the Laplace-Beltrami operator on a manifold $$M$$, i.e.

$$-\Delta\phi_\lambda = \lambda\phi,$$

then as $$\lambda \to \infty$$, there are constants $$c$$, $$C$$ such that

$$c\sqrt\lambda \le \text{area}(\{\phi_\lambda = 0\}) \le C\sqrt\lambda.$$

I've seen several results for manifolds without boundary of varying degrees of regularity. For example, Donnelly and Feffermann proved that Yau's conjecture is true for real analytic manifolds, but as I understand, the conjecture hasn't been proven yet for $$C^\infty$$ manifolds.

Are there extensions of the Yau conjecture to manifolds with boundary? Most of the work I've found (mainly other papers by Malinnikova and Logunov) considers only manifolds without boundary.

• Is it fair to assume you are interested in Dirichlet/Neumann boundary conditions? Oct 22, 2019 at 22:13

Theorem 1.2 On any real analytic Riemannian manifold $$M$$ with boundary, the $$n-1$$ dimensional Hausdorff measure, $$\mathcal{H}^{n-1}(N)$$, of the nodal set $$N$$ of eigenfunction $$f$$, $$\Delta f = -\lambda f$$, (with Dirichlet or Neumann conditions on the boundary of $$M$$) satisfies $$c_1\sqrt{\lambda}\leq \mathcal{H}^{n-1}(N) \leq c_2 \sqrt{\lambda},$$ for some positive constants $$c_1$$, $$c_2$$.
• Theorem 1.2 in that paper is only for real analytic manifolds. Theorem 1.1, which applies to $C^\infty$ manifolds, only says that eigenfunctions vanish to order at most $c\sqrt{\lambda}$. Apr 16, 2020 at 10:21