 97445 HoffmannOstenhof M., HoffmannOstenhof T., Nadirashvili, N.
 Critical Sets of Smooth Solutions to Elliptic Equations
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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
$n2$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
$n2$dimensional Hausdorff measure.
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