Phase boundary
The behavior of phase boundaries has been a developing subject of interest and an active research field in physics and mathematics for almost two centuries. One reason behind this is that phase boundaries naturally arise in many physical processes due to immiscibility of two or more substances with different physical properties. Hence, various phenomena such as capillarity effect, growth of grain boundaries, physics of binary alloys, formation of snow flakes fall under the category of interface science.
One of the oldest problems in the area dates back to Lame and Clapeyron [1] who studied the freezing of the ground. Their goal was to determine the thickness of solid crust generated by the cooling of a liquid at constant temperature filling the half-space. In 1889, Stefan, while working on the freezing of the ground developed these ideas further and formulated the two phase model which came to be known as the Stefan Problem.[2]
The proof of existence and uniqueness of a solution to the Stefan problem was done in many stages. Proving the general existence of the solutions turned out to be a difficult problem for 
(see e.g. \cite{3}) that was finally solved by Enverbek Meirmenov.[3]