Uma solução do problema isoperimétrico usando séries de Fourier.

Abstract

Isoperimetric problem is a theme whose first appearance is contained in the work Aeneid, written before Christ, and which can also be noted in the shape of the oldest cities core emphasizing the fact that the ancestors knew its solution. The problem in its origin is to consider a length and find among all closed curves of the plane which one contains the largest area. It has several applications, such as the construction of fences and packages. This research is characterized as a bibliographic review of books, articles and monographs, whose objective is to present a statement of isoperimetric inequality on the plane through the Fourier series, and expose some of its applications. In this regard, results and definitions about Fourier series and theory of plane curves were presented in order to facilitate its understanding. Isoperimetric inequality appears as a relevant subject for trainee mathematicians, since it can be introduced into basic education through didactic sequences. Nevertheless, its resolution in the plane involve a significant part of mathematics presented in higher education, and furthermore this problem is the aim of researches in several fields of mathematics, such as variational calculus and Banach space theory. Therefore, Isoperimetric Problem establishes a link between mathematics at different levels, promoting a significance for such study.