Compactness theorems for the study of two-phase Stefan type problems

In standard compactness theorems for functions from Sobolev spaces with integer exponents, compactness of a set in the $W^k_p$ norms usually requires its uniform boundedness in the space $W^{k+1}_{p_1}$. The paper considers the case (for $k=1$) when there are no uniform estimates of second derivatives throughout the entire domain of definition. However, they exist for sequences of subregions, each of which is determined by its own sequence of curves (in the flat case), which approach each other as the number increases.
The need for such theorems arises in the study of multiphase Stefan problems, in which such a situation is observed when constructing approximate solutions. These results make it possible to make limit transitions using approximate solutions in two-phase problems with an unknown boundary, which describe the processes of transition of matter from one state to another.