Calculus of Variations

By Michael Fotopoulos, Monash University

Most mathematicians and mathematics appreciators alike would be familiar with the broad and quintessential field of calculus. Although not properly formalised until the 1800’s by the efforts of Cauchy and Weierstrass, calculus has been instrumental in the development of physics, mathematics, engineering, and even biology and chemistry from the 1600’s right through to the modern day.

To briefly refresh the reader, calculus concerns itself with extrema of mathematical objects – be it ordinary functions, surfaces, or in the case of my research, functionals. Naïvely speaking a functional is like a function that takes functions as its parameters. Generally, locating and classifying extrema involves taking derivatives but unfortunately this is not so simple for functionals. This precise dilemma led to the development of the Calculus of Variations. Without inciting too much technical jargon and definitions – calculus of variations allows us to devise a rigorous definition for these “functional derivatives” and perform mathematics with them.

My research concerned itself with finding extrema of a particular functional (hereafter called an “energy”) that one can attribute to two-dimensional surfaces living in three-dimensional Euclidian space. To understand more precisely the energy with which I interested myself we must journey now to another field of mathematics called differential geometry. The element of this field with which I concerned myself was mean curvature. You can think of curvature as a measure of how much a surface deviates from a plane; a point with positive mean curvature would bend down all around like the top of a mountain, and vice-versa a point with negative mean curvature would bend up to form a bowl or a valley. From this notion of mean curvature, we can define an energy that computes the total mean curvature of a given surface (via an integral over the entire surface). Surfaces that minimise this energy, we appropriately call minimal surfaces, whose mean curvature is 0 everywhere. Examples of these surfaces are the shapes formed by soap films when stretched between two solid surfaces. There are many other examples of minimal surfaces whose properties have been well studied in the literature for some time.

For me specifically, I studied a slightly different energy (whose formal definition is too unwieldy for the scope of this blog) whose minimisers form a larger class of surfaces called “constant mean curvature” surfaces. Obviously, these surfaces also contain the above described minimal surfaces for which the mean curvature is the constant 0. In my report, I detail a method for uncovering a system of partial differential equations whose solutions are not only these types of surfaces as well as many more who describe other extrema of the energy in question.

As such a broad topic, this research project allowed me to learn and utilise a number of different fields of mathematics, laying the foundations for more in depth future work and research in analysis.

 

Michael Fotopoulos was one of the recipients of a 2017/18 AMSI Vacation Research Scholarship.

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