On extremum sufficient conditions in multidimensional variation calculus problems

Variation principles give formalization and general approach to construct and study the models from different fields of knowledge. They provide system presentations about theories origins. In the models, sought-for solution is a stationary point of a criterion. Its search on the basis of necessary conditions should not accomplish problem investigation. Sufficient conditions are needed to assert its optimality. In natural sciences, such results substantiate principles of energy or Hamilton's action minimization. In the paper, invariant surface integrals discovery gave possibility to prove minimum availability in multidimensional variation calculus problems. The functional in the problems may depend on several unknown functions of many variables and their high-order derivatives. Classical theorems are generalized.