On the Betti numbers of level sets of solutions to elliptic equations

In this paper we study the topological properties of the level sets, §t(u) = (x: u(x) = t), of solutions u of second order elliptic equations with vanishing zeroth order terms. We show that the total Betti number of level sets §t is a uniformly bounded function of the parameter t. The uniform bound can be estimated in terms of the analytic coefficients as well as the generalized degrees of the corresponding solutions. Such an estimate is also valid for the nodal sets of solutions of the same type equations with zeroth order terms. In general, it is possible to derive from our analysis an estimate for the total Betti numbers of level sets, for large measure set of t′ s, when coefficients are sufficiently smooth, and therefore a L^p bound on Betti numbers as a function of t. These estimates are obtained by a quantitative Stability Lemma and a quantitative Morse Lemma.