Isoperimetric Inequalities for the Stefan Problem
Björn Gustafsson, Jacqueline Mossino · SIAM Journal on Mathematical Analysis · 1989
The weak solution $(\theta ,h)$ of the Stefan problem in some annular domain $\omega \times (0,T)$ is compared with the weak solution $(\Theta ,H)$ of the “symmetrized” problem, in $\Omega \times (0,T)$ where $\Omega $ is a symmetrical annulus having the same measure as $\omega $. For the one-phase Stefan problem—$\theta \geqq 0$, $h \in ( - \alpha ,0)$ when $\theta = 0$—it is shown in particular that the “volume of ice” (meas $\{ {h(t) = - \alpha } \}$) remains greatest in spherical symmetry (with initial data decreasing along the radii).