Critical Sets of Elliptic Equations

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 ( u )u { x :|δ u |( x ) = 0}, as well as the first estimates on the effective critical set r ( u ), which roughly consists of points x such that the gradient of u is small somewhere on B r ( x ) compared to the nonconstancy of u . The results are new even for harmonic functions on ℝ n . Given such a u , the standard first‐order stratification { l k } 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{l n , r k}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{l n , r k} , 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.