Parabolic boundary value problems connected with Newton’s polygon and some problems of crystallization

. A new class of boundary value problems for parabolic operators is introduced which is based on the Newton polygon method. We show unique solvability and a priori estimates in corresponding L 2 -Sobolev spaces. As an application, we discuss some linearized free boundary problems arising in crystallization theory which do not satisfy the classical parabolicity condition. It is shown that these belong to the new class of parabolic boundary value problems, and two-sided estimates for their solutions are obtained.