ABSTRACT. In this paper we study the regularity of stationary and minimizing harmonic maps f : B 2 (p) ⊆ M → N between Riemannian manifolds. If S k ( f ) ≡ {x ∈ M : no tangent map at x is k + 1-symmetric} is the k th -stratum of the singular set of f , then it is well known that dim S k ≤ k, however little else about the structure of S k ( f ) is understood in any generality. Our first result is for a general stationary harmonic map, where weIn the case of minimizing harmonic maps we go further, and prove that the singular set S ( f ), which is well known to satisfy dim S ( f ) ≤ n − 3, is in fact n − 3-rectifiable with uniformly finite n − 3-measure. An effective version of this allows us to prove that |∇ f | has estimates in L Reifenberg, and give checkable criteria to determine when a set is k-rectifiable with uniform measure estimates.The new Reifenberg type theorems may be of some independent interest. The L 2 -subspace approximation theorem we prove is then used to help break down the quantitative stratifications into pieces which satisfy these criteria.

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Given a solution u to a linear, homogeneous, second‐order elliptic equation with Lipschitz coefficients, we introduce techniques for giving improved estimates of the critical set 𝒷(u)u {x :|δu|(x) = 0}, as well as the first estimates on the effective critical set 𝒷r(u), which roughly consists of points x such that the gradient of u is small somewhere on Br(x) compared to the nonconstancy of u. The results are new even for harmonic functions on ℝn. Given such a u, the standard first‐order stratification {lk} of u separates points x based on the degrees of symmetry of the leading‐order polynomial of u‐u(x). In this paper we give a quantitative stratification { ln,rk } of u, which separates points based on the number of almost symmetries of approximate leading‐order polynomials of u at various scales. We prove effective estimates on the volume of the tubular neighborhood of each { ln,rk }, which lead directly to (n‐2 + ɛ)‐Minkowski type estimates for the critical set of u. With some additional regularity assumptions on the coefficients of the equation, we refine the estimate to give new proofs of the uniform (n‐2)‐Hausdorff measure estimate on the critical set and singular sets of u.© 2014 Wiley Periodicals, Inc.

We complete the picture of sharp eigenvalue estimates for the p-Laplacian on a compact manifold by providing sharp estimates on the first nonzero eigenvalue of the nonlinear operator ∆ p when the Ricci curvature is bounded from below by a negative constant. We assume that the boundary of the manifold is convex, and put Neumann boundary conditions on it. The proof is based on a refined gradient comparison technique and a careful analysis of the underlying model spaces.

In this paper we study solutions to elliptic linear equations L(u) = ∂ i (a i j (x)∂ j u) + b i (x)∂ i u + c(x)u = 0, either on R n or a Riemannian manifold, under the assumption that the coefficient functions a i j are Lipschitz bounded. We focus our attention on the critical set C(u) ≡ {x : |∇u| = 0} and the singular set S(u) ≡ {x : u = |∇u| = 0}, 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 H n−2 (S) < ∞ unless the coefficients are smooth. Fundamentally, this is due to the need of an ǫ-regularity theorem which 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 ǫ-regularity. Consequently, we prove that if the frequency of u is bounded by Λ, then we have the estimates, 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.