Higher-Dimensional Calculus of Variations

Abstract

In this chapter, we discuss the main ideas of the higher-dimensional calculus of variations, the theory of minimization of a multiple integral. The minimization problem for a multiple integral arises in various fields of the natural sciences. For example, various problems in continuous medium theory (arising in hydrodynamics and gas dynamics, elasticity and plasticity theory, shell theory, etc.) lead to the minimization problem for a multiple integral when applying variational principles. Variational principles are especially important in quantum field theory because this approach is practically the only way to obtain equations describing the evolution of a quantum system. We begin with one of the main problems of the higher-dimensional calculus of variations, the problem of minimal surfaces.