Let ∆M be the Laplace operator on a compact n-dimensional Riemannian manifold without boundary. We study the zero sets of its eigenfunctions u : ∆u + λu = 0. In dimension n = 2 we refine the Donnelly-Fefferman estimate by showing that H 1 ({u = 0}) ≤ Cλ 3/4−β , β ∈ (0, 1/4). The proof employs the Donnelli-Fefferman estimate and a combinatorial argument, which also gives a lower (non-sharp) bound in dimension n = 3: H 2 ({u = 0}) ≥ cλ α , α ∈ (0, 1/2). The positive constants c, C depend on the manifold, α and β are universal.

Let u be a solution to an elliptic equation div(A∇u) = 0 with Lipschitz coefficients in R n . Assume |u| is bounded by 1 in the ball B = {|x| ≤ 1}. We show that if |u| < ε on a set E ⊂ 1 2 B with positive n-dimensional Hausdorf measure, thenwhere C > 0, γ ∈ (0, 1) do not depend on u and depend only on A and the measure of E. We specify the dependence on the measure of E in the form of the Remez type inequality. Similar estimate holds for sets E with Hausdorff dimension bigger than n − 1.For the gradients of the solutions we show that a similar propagation of smallness holds for sets of Hausdorff dimension bigger than n − 1 − c, where c > 0 is a small numerical constant depending on the dimension only.

This paper focuses on a relation between the growth of harmonic functions and the Hausdorff measure of their zero sets. Let u be a real-valued harmonic function in R n with u(0) = 0 and n ≥ 3. We provewhere the doubling index N is a notion of growth defined by sup|u|.This gives an almost sharp lower bound for the Hausdorff measure of the zero set of u, which is conjectured to be linear in N . The new ingredients of the article are the notion of stable growth, and a multi-scale induction technique for a lower bound for the distribution of the doubling index of harmonic functions. It gives a significant improvement over the previous best-known boundWe often write N (x, r) instead of N h (B(x, r)) and often omit the dependence on h in the notation and simply write N (B). Theorem 1.1 ([4], [6]). Let B ⊂ R n be a unit ball. There exists a constant C = C(n) > 1 such that H n−1 ({u = 0} ∩ B) ≤ CN (B) , 1