Volume Estimates on the Critical Sets of Solutions to Elliptic PDEs

In this paper we study solutions to elliptic linear equations (Formula presented.) either on (Formula presented.) or a Riemannian manifold, under the assumption that the coefficient functions aij are Lipschitz bounded. We focus our attention on the critical set (Formula presented.) and the singular set (Formula presented.), and more importantly on effective versions of these. Currently, with just the Lipschitz regularity of the coefficients, the strongest results in the literature say that the singular set is (n–2)–dimensional; however, at this point it has not even been shown that (Formula presented.) unless the coefficients are smooth. Fundamentally, this is due to the need of an ɛ-regularity theorem that requires higher smoothness of the coefficients as the frequency increases. We introduce new techniques for estimating the critical and singular set, which avoids the need of any such ɛ-regularity. Consequently, we prove that if the frequency of u is bounded by Λ, then we have the estimates (Formula presented.) and (Formula presented.), depending on whether the equation is critical or not. More importantly, we prove corresponding estimates for the effective critical and singular sets. Even under the assumption of real analytic coefficients these results are much sharper than those currently in the literature. We also give applications of the technique to give estimates on the volume of the nodal set of solutions and estimates for the corresponding eigenvalue problem.