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

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