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.