The intertwined symphony

Calculus of variations and partial differential equations (PDEs) may seem like abstract mathematical concepts, but they play a fundamental role in understanding and improving many  aspects of  science and engineering. Among other things, they help in designing  bridges or understanding  the behaviour of light.

Let’s imagine building a bridge between two points. There are countless ways to achieve this, but some designs will be more efficient than others. Calculus of variations helps us to find the optimum design by figuring out  which shape of the bridge will use the minimal amount of a given material. It provides a powerful framework for seeking the “best” solution among a collection of possibilities.

Partial differential equations, on the other hand, describe how things (variables) change when there are multiple factors involved.  As we will show later in the text, they are crucial in modeling various phenomena where many factors are at play .

The beauty lies in the deep connection between calculus of variations and PDEs. Many physical principles can be seen as minimization problems, leading to specific PDEs via the Euler-Lagrange equations. For example, if you want to move an object from point A to point B using the smoothest or “easiest” way possible (like saving energy or time), then the Euler-Lagrange equations are your tool. . To put it in more general technical terms,  solutions to certain PDEs can be found by interpreting them as minimizers of a well-defined functional. This interplay allows mathematicians and scientists to use the strengths of both fields to tackle complex problems.

Dr. Rémy Rodiac’s research project focuses on developing new techniques in calculus of variations and PDEs where, in spite of many results already obtained, several challenging questions remain open.