Compactness theorems connected with problems with unknown boundary

The paper provides some of the results that were presented at the IX International Conference on Mathematical Modeling dedicated to the 75th anniversary of V. N. Vragov and are related to the study of problems with an unknown boundary by methods of compactness. A substantiation of the theorem on relative compactness is given, which can be used in the study of problems of the Stefan type with an unknown part of the boundary, as well as in problems for equations of variable type with an unknown boundary of type change. An example of such problem is given, and it is shown that the estimates obtained for approximate solutions of the equation and functions describing a sequence of boundaries approaching the sought boundary of the phase transition completely coincide with the conditions formulated in the main theorem on compactness.
The author is not aware of theorems of such type. The theorem is a new kind of compactness theorem, adapted to problems of the Stefan type.
For better perception, the simplest conditions are given under which the result is valid, which coincide with the conditions of the considered example. However, the result can be generalized to much more general situations, including the number of phase transition boundaries and replacing the estimate of the second derivative with an estimate of a more complex aggregate that occurs in equations with degenerations on solutions.