97-445 Hoffmann-Ostenhof M., Hoffmann-Ostenhof T., Nadirashvili, N.

Critical Sets of Smooth Solutions to Elliptic Equations (434K, Postscript) Aug 13, 97

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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.

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