Calculus of Variations

Abstract

The origins of “Calculus of Variations” can be traced back to the works of Aristotle (384–322 BC) and Zenodotos of Ephessus (fl. 280 BC). Calculus of variations is an extension of calculus and so, it is not surprising that it has found important applications in many other sciences. Indeed, the macroscopic statements afforded by variational principles may provide the only valid mathematical formulation of many physical laws. Moreover, the calculus of variations in the early fifties “gave birth” to the theory of optimal control, in order to deal with new applications coming from aerospace engineering, industrial process control and mathematical economics. In fact, the emergence of optimal control theory revitalized the subject of calculus of variations, which continued to develop vigorously, in particular in the direction of multiple integral problems, where important breakthroughs were made both in theory and applications (nonlinear elasticity). The aim of this chapter is to highlight the important aspects of this theory. A basic method of this theory is the so-called “direct method”, according to which we deal with directly with the functional J which is to be minimized in a function space. The idea of the direct method is to find minimizing sequences which belong to a bounded and closed set and to ensure that J is lower semicontinuous. Due to the infinite dimensional nature of the problem, this is a delicate process. It requires the use of a topology which is weaker than the usual metric topology, with which a version of the Heine-Borel theorem applies. This can be achieved using classical results of functional analysis such as the theorem of Alaoglu and the Eberlein-Smulian theorem. The second requirement of lower semicontinuity of J is more difficult to satisfy and is linked to some kind of convexity of the integrand function. This is perhaps not surprising if we recall Mazur’s lemma, which illustrates that the algebraic notion of convexity bridges the gap between strong and weak topologies.

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The erratum of this chapter is available at http://dx.doi.org/10.1007/978-1-4615-4665-8_13