Hoffmann-Ostenhof M., Hoffmann-Ostenhof T., Nadirashvili, N.
Critical Sets of Smooth Solutions to Elliptic Equations
(434K, Postscript)
ABSTRACT. Let $u\not\equiv\text{const}$ satisfy an elliptic equation
$L_0u\equiv\Si a_{i,j}D_{ij}u+\Si b_jD_j u=0$ with smooth
coefficients in a domain in $\Bbb R^n$. It is shown that the
critical set $|\nabla u|^{-1}\{0\}$ has locally finite
$n-2$-dimensional Hausdorff measure. This implies in
particular that for a solution $u\not\equiv0$ of $(L_0+c)u=0$,
with $c\in C^\infty$, the critical zero set
$u^{-1}\{0\}\cap|\nabla u|^{-1}\{0\}$ has locally finite
$n-2$-dimensional Hausdorff measure.