Metric properties of minimal solutions of discrete periodical variational problems

In this paper we investigate properties of minimal solutions of multidimensional discrete periodical variational problems. A one-dimensional example of such a problem is the Frenkel-Kontorova model. We pick out a family of self-conformed solutions, properties of which are exactly the same as in the one-dimensional case. We investigate also non-self-conformed solutions. For translationally invariant Lagrangians we prove that only self-conformed solutions are physically practicable.