Critical sets of solutions to elliptic equations

Abstract

Let u ≢ const, satisfy an elliptic equation ℒ0u ≡ ΣaijDiju + ΣbjDju = 0 with smooth coefficients in a domain in Rn . It is shown that the critical set |∇u|−1(0) has locally finite (n − 2)-dimensional Hausdorff measure. This implies in particular that for a solution u ≢ 0 of (ℒ0 + c)u = 0, with c ∈ C∞, the singular set uℒ1(0) ∩ |∇u|−1(0) has locally finite (n ℒ 2)-dimensional Hausdorff measure. © 1999 J. differential geometry.