I see in this case a surprising and pleasing adequacy of a very simple modelling with reality. “I like the fact that the minimal surfaces I'm studying do indeed resemble the soap films of everyday life. But curiously, the one I prefer and which seems to me to best describe practical soap films, the theorem of the existence of minimisers is not yet known, even for surfaces of dimension 2 in Euclidean space of dimension 3, with an edge that is a smooth curve,” he said. “In fact, there are many ways to give precise meanings to the terms “leaning” and “surface area”, which have resulted in some wonderful pages of mathematics. “It's called Plateau’s problem, which is precisely about having surfaces that rest on a given edge (the wire) and a minimum area,” researcher Mr David explains. In this case, the model consists in saying that these films and bubbles minimise a very simple quantity, the total area, with the constraint of leaning on a given edge, which corresponds to dipping a wire in soap and looking at the surface formed (or containing given volumes in the bubbles). A very good example, studied by Guy David, is the description of soap films and bubbles. And then to study these minimal configurations in detail. Many phenomena are governed by the minimisation of “functionals”, and the main goals of the calculus of variations are to demonstrate the existence of functions, or configurations, that minimise these functionals. What do they most often look like? On plans? With what error estimates? What can we deduce about all kinds of objects associated with these sets? It is also one of the main tools of the “calculus of variations”. One of the useful tools for the study of harmonic measure is the so-called geometric measure theory, a way to study in detail the regularity of sets in Euclidean space. The study of harmonic measure proved to be fascinating here, and also allowed a better understanding of certain problems that remained unaddressed in the classical case of Laplace's equation. The Brownian motion analogue is thus pushed towards the edge (otherwise it could pass by without seeing it!). With two other researchers, Guy David has introduced a different class of elliptic operators, adapted to domains with a much smaller boundary. Significant progress has recently been made. The way air moves (Brownian motion) is then governed by a simple partial differential equation, Laplace’s equation, which is a good model of what are called elliptic equations,” he says. In this case, the so-called harmonic measure describes the part of the lung that can be reached by air. “The question is whether the air, whose movement is described by a Brownian motion, can reach all parts of the lung or only a small part. “The lung is made to have the largest possible surface area, to allow maximum exchange between oxygen and carbon dioxide, and therefore has a multitude of cavities,” Guy David explains. ![]() A small particle of air enters it and the question is how far it reaches inside the lung. The study of the relationships between the geometry of a domain (the inside of the lung) and harmonic measure (the distribution of air at the interface) is one of the great classics of partial differential equations. If we know the air pressure on Earth at a given time, then we should be able to tell the pressures for all other times.” From there, mathematicians ask themselves three questions: Do their equations describe the nature around them? Will they be able to solve them? And once these two steps have been taken, will they also be able to resolve them practically? Harmonic measure “It is common practice, for example, to describe the weather with variables such as air pressure or temperature. “Partial differential equations represent the way we model nature,” Guy David says. The French Academy of Sciences has just awarded him the 2020 Ampère prize. ![]() Today, he is mainly interested in a concept of harmonic measure adapted to domains with small edges. ![]() His recent research concerns, for example, the theoretical description of soap films. ![]() He teaches in the Mathematics Department of the University. He is studying the geometric theory of measure, calculus of variations and partial differential equations. Guy David is a teacher and researcher at the Orsay mathematics laboratory (LMO - Université Paris-Saclay, CNRS).
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