Hardt R., Hoffmann-Ostenhof M., Hoffmann-Ostenhof T., Nadirashvili N.
Critical sets of solutions to elliptic equations
(34K, LaTeX2e)

ABSTRACT.  Let $u\not\equiv\operatorname{const}$ satisfy an
elliptic equation $L_0u\equiv\sum a_{ij}D_{ij}u+\sum b_jD_ju=0$ with
smooth coefficients in a domain in $\mathbf R^n$. 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.