I've just read a [review paper](https://arxiv.org/pdf/1908.01639.pdf) 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.



Yau's conjecture on nodal sets for manifolds with boundary
Yau's conjecture was proved for the real analytic manifolds by Donelly and Fefferman, first for manifolds without boundary, then for manifolds with [boundary](https://www.sciencedirect.com/science/article/pii/B9780125742498500171) in the paper *Nodal Sets of Eigenfunctions: Riemannian Manifolds With Boundary*, Analysis, et Cetera (1990), 251--262. The main result there is stated below.



> 
> **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 $c\_1$, $c\_2$.
> 
> 
> 


There are other papers that handle more general boundary conditions, many of which reference the above paper.


