As picture below ,how to compute the $\partial_t |\Omega_t|$ ? The picture below is from the 32 page of Maximum principles and the method of moving planes.
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2026-03-28 07:15:49.1774682149
Derivative of volume of given set
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Use the Co-area formula with $g=|\nabla u|^{-1} \chi_{\Omega_t}$ to get
$$ \int_{\Omega_t} dA=\int_t^\infty \left(\int_{\partial \Omega_x} \frac{ds}{|\nabla u|}\right)dx.$$
Differentiating this with respect to $t$ gives the formula you're looking for.