Existence and invertibility theorems for the problem of the variational construction of quasi-isometric mappings with free boundaries

Earlier, some existence and invertibility theorems were proved for the problem of construction of multidimensional quasi-isometric mappings as minimizers of a polyconvex functional. In this paper, the proof of the same theorems is given that uses only natural assumptions on the shape of the domains and does not rely on additional a priori considerations. The general scheme of the proof also covers the situations in which the mapping of the entire or part of the domain boundary is found by minimizing the functional. The proof is based on the techniques used in the theory of the existence of minima in polyconvex problems.