Calculus of Variations

The problem of finding the maximum value or minimum value of a function over some set in the domain of the function is called: Variational problemvariational problem. This problem and finding ways to deal with it is as old as humanity itself. In case we are looking for a minimum value, the problem is called: minimization problem. We will focus on the minimization problems due to their particular importance: In physics and engineering, we look for the minimum energy, in geometry, we look for the shortest distance or the smallest area or volume, and in economy, we look for the minimum costs. Historically, the minimization problems in physics and geometry were the main motivation to develop the necessary tools and techniques that laid the foundations of this field of mathematics which connects functional analysis with applied mathematics.