Let $(\phi_k,\lambda_k)$ be the couple of eigenfunctions and eigenvalues of the the Laplacian operator on $\Omega \subset \mathbb R^n$. The nodal set of $\phi_k$ is the set $$\mathcal N_k = \{x \in \Omega: \phi_k(x) = 0\}.$$ The nodal domains of $\phi_k$ are the connected components of $\Omega \setminus \mathcal N_k$.
If $\Omega = [0,1]$, it's simple to check that $$\phi_k = \sqrt{2} \sin(k \pi x)$$ and $$\mathcal N_k = \{x = i/k: i = 1,\dots, k-1\}$$ and therefore the nodal domains of $\phi_k$ are $k$ intervals of equal length.
Does something similar hold if we take $\Omega$ to be a smooth (connected) domain of $\mathbb R^n$ with $n>1$? In particular, do the nodal domains of $\phi_k$ have the same measure? Can we say something about their shape? ... What is known about this topic?
