  This paper is concerned with the location of nodal sets of eigenfunctions of
the Dirichlet Laplacian in thin tubular neighbourhoods of hypersurfaces of the
Euclidean space of arbitrary dimension. In the limit when the radius of the
neighbourhood tends to zero, it is known that spectral properties of the
Laplacian are approximated well by an effective Schr\"odinger operator on the
hypersurface with a potential expressed solely in terms of principal
curvatures. By applying techniques of elliptic partial differential equations,
we strengthen the known perturbation results to get a convergence of
eigenfunctions in H\"older spaces. This enables us in particular to conclude
that every nodal set has a non-empty intersection with the boundary of the
tubular neighbourhood.
