Critical sets of solutions to elliptic equations

R. Hardt, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, N. Nadirashvili
1999 Journal of differential geometry  
Let u 6 const satisfy an elliptic equation L 0 u P a ij D ij u + P b j D j u = 0 with smooth coe cients in a domain in R n . It is shown that the critical set jruj ,1 f0g has locally nite n , 2 dimensional Hausdor measure. This implies in particular that for a solution u 6 0 of L 0 + cu = 0 , with c 2 C 1 , the singular set u ,1 f0g j r uj ,1 f0g has locally nite n , 2 dimensional Hausdor measure.
doi:10.4310/jdg/1214425070 fatcat:ayk63njbz5cephe4lxpsmaqb2y