An existence theorem and weak Lagrange equations for a variational problem of the theory of phase transitions
The model of phase transitions in continuous media according to which the total energy of a system consists of the sum of energies of deformation of each of the phases and the area of the interface is considered. For that model an existence theorem for a global minimum and weak Lagrange equations for the energy functional are investigated. To overcome the difficulty connected with the rather complicated structure of the area of the interface between the phases, the perimeter of the set occupied by one of the phases is applied. If the set has a sufficiently smooth boundary then its perimeter coincides with the area of the interface. It is also proved that under additional smoothness the weak Lagrange equations transform into the classical Lagrange equations and a system of boundary conditions on the interface.
scientific article; zbMATH DE number 917095