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We consider a two-phase Stefan problem for a semi-infinite body x>0, with a convective boundary condition including a density jump at the free boundary with a time-dependent heat transfer coefficient of the type h/t, h>0 whose solution was given in D. A. Tarzia, PAMM. Proc. Appl. Math. Mech. 7, 1040307-1040308 (2007). We demonstrate that the solution to this problem converges to the solution to the analogous one with a temperature boundary condition when the heat transfer coefficient h ->+infinity. Moreover, we analyze the dependence of the free boundary respecting to the jump density.

We study the thin-shell conjecture for the Schatten classes. In particular, we establish the conjecture for the operator norm, and we also improve on the best known bound for the Schatten classes, due to Barthe and Cordero-Erausquin (Proc. Lond. Math. Soc. 106 (2013) 33-64) or Lee and Vempala (2017), for a few more cases. We also show that a necessary condition for the conjecture to be true for any of the Schatten classes is a rather strong negative correlation property: as a consequence of this we obtain the validity of this negative correlation property for all the cases for which we already know the conjecture is true (as for example for the operator norm), but moreover also for all the cases for which we can get a better estimate than the one in (Proc. Lond. Math. Soc. 106 (2013) 33-64) or (Lee and Vempala (2017)). For the proofs, our starting point is techniques that were employed for the Schatten classes in (Math. Ann. 312 (1998) 773-783) and (Ann. Inst. Henri Poincare Probab. Stat. 43 (2007) 87-99) with regard to other problems.