On the Existence of Approximate Solutions of Variational Problems in Nonlinear Elasticity Theory

The article is devoted to the substantiation of approximate methods for solving problems in the nonlinear theory of elasticity. The variational approach proposed by J. Ball is used, in which the solution to the problem of determining the shape of a deformed body is reduced to solving the corresponding variational problem for the minimum of the stored energy functional. In this case, the specific form of this functional is specified by the type of elastic material and is written in integral form. In this article, a construction of an approximate solution is proposed using the Delaunay triangulation of a polygonal domain in the class of piecewise linear nondegenerate mappings. The article introduces a class of mappings admitting such an approximation. It is proved that the constructed piecewise linear mappings form a minimizing sequence for the stored energy functional. Also, the article finds conditions under which this sequence converges to the exact solution of the original variational problem in a suitable class of mappings. The case of functionals with linear growth is considered separately - an integral inequality is obtained that ensures the existence of an approximate solution. It is noted that similar conditions naturally arise for area-type functionals in problems of the existence of capillary surfaces and surfaces with a prescribed mean curvature.