The sharp upper bound for the area of the nodal sets of Dirichlet Laplace eigenfunctions

Let $$\Omega $$ be a bounded domain in $${\mathbb {R}}^n$$ with $$C^{1}$$ boundary and let $$u_\lambda $$ be a Dirichlet Laplace eigenfunction in $$\Omega $$ with eigenvalue $$\lambda $$ . We show that the $$(n-1)$$ -dimensional Hausdorff measure of the zero set of $$u_\lambda $$ does not exceed $$C(\Omega )\sqrt{\lambda }$$ . This result is new even for the case of domains with $$C^\infty $$ -smooth boundary.