Critical sets of solutions to elliptic equations
Robert Hardt, Maria Hoffmann-Ostenhof, Thomas Hoffmann-Ostenhof, Nikolaï Nadirashvili · Journal of Differential Geometry · 1999
Let u ^ const, satisfy an elliptic equation CQU = Yl o-ijDijU + Yl bjDJU = 0 with smooth coefficients in a domain in R n .It is shown that the critical set |Vu| -1 {0} has locally finite (n -2)-dimensional Hausdorff measure.This implies in particular that for a solution u ^ 0 of (CQ + C)U = 0, with c G C°°, the singular set u -1 {0} n |V«| -1 {0} has locally finite (n -2)-dimensional Hausdorff measure.