98-128 Hardt R., Hoffmann-Ostenhof M., Hoffmann-Ostenhof T., Nadirashvili N.

Critical sets of solutions to elliptic equations (34K, LaTeX2e) Mar 4, 98

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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.

Files: 98-128.tex