Corner formation for the undercooled Stefan problem (Q2706073)

This paper contains an extensive analysis of the development of corners in the free boundary for the one-phase undercooled Stefan problem in two and three space dimensions. A concise review of the studies of the singularities of the Stefan problem is presented in the introduction. The scope is not only to summarize the state of the art, but above all to emphasize the genesis of singularities as well as to illustrate the main tools in such studies and the reasons for their success. Section 2 describes the general approach used in the paper, Section 3 is split in two parts devoted to the two-dimensional case and to the three-dimensional core respectively. The subjects dealt with are: how singularities are produced, what is their shape and what the behaviour of the solution near them. The theory presented is rather complex and rich. After having shown how singularities are formed, say at time \(t=0\), the author addresses the question of the asymptotical behaviour of the solutions for \(t>0\) (Section 4), i.e., how they evolve from the onset of the singularity. In all the situations considered many different cases are possible and the corresponding analysis is full of technical details. For this reason the paper is supplemented by three large appendices dealing with some specific topics.