  Let $u$ be a harmonic function in the unit ball $B(0,1) \subset
\mathbb{R}^n$, $n \geq 3$, such that $u(0)=0$. Nadirashvili conjectured that
there exists a positive constant $c$, depending on the dimension $n$ only, such
that $H^{n-1}(\{u=0 \}\cap B) \geq c$. We prove Nadirashvili's conjecture as
well as its counterpart on $C^\infty$-smooth Riemannian manifolds. The latter
yields the lower bound in Yau's conjecture. Namely, we show that for any
compact $C^\infty$-smooth Riemannian manifold $M$ (without boundary) of
dimension $n$ there exists $c>0$ such that for any Laplace eigenfunction
$\varphi_\lambda$ on $M$, which corresponds to the eigenvalue $\lambda$, the
following inequality holds: $c \sqrt \lambda \leq H^{n-1}(\{\varphi_\lambda
=0\})$.
