![]() ![]() 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. If m = 2 n the Plateau curves become a circle with a center at (1, 0) and a radius of 2.Guy David is a teacher and researcher at the Orsay mathematics laboratory (LMO - Université Paris-Saclay, CNRS). Y = 2 a sin( mt ) sin( nt ) / sin( m – n) t. That are described by the parametric equations: Plateau curves are a set of curves, named after Joseph Plateau, Princeton, NJ: Princeton University Press, 1989. Plateau's Problem and the Calculus of Variations. With its corners chopped off or the bottom half of a cone mounted on a cylinder,Īre known as Wullf shapes, and provide fertile ground forġ. ![]() Minimal surfaces that model these conditions, like a cube In a piece of wood), but they still require the least energy to encloseĪ given volume. In crystals this is not the case (magnitudes of surface forces differ inĭifferent directions, though they may exhibit a grain, analogous to that Surface of a soap bubble all have the same magnitude in all directions. That had been open for more than a century. Yet unobserved configurations were possible – thus settling a question To her doctoral thesis, Taylor was able to prove that Plateau's rules wereĪ necessary consequence of the energy-minimizing principle – no other Plateau's patterns were just a set of empirical rules. Until the American mathematician Jean Taylor came along in the mid-1970s, When dipped in a soapy solution, form angles of roughly 109° at a central On the other hand, theĮdges of the six soap-film faces that emerge within a tetrahedral wire frame, Outer surfaces of the bubbles in 120° angles. Sizes) will have a common dividing wall (the third surface) that meets the In a cluster of bubbles, two intersecting bubbles (of possibly different Six surfaces meet at a vertex, forming angles of about 109°. Two ways: either three surfaces meet at 120° angles along a curve or Plateau claimed that soap bubble surfaces always make contact in one of The geometry of how soap bubbles fit together. Plateau noticed that a handful of simple patterns seemed to completely describe Plateau's problem is the general problem of determining the shape of the minimal Looking at the Sun for 25 seconds: this damaged his eyes, and he eventually In 1829 Plateau carried out an optical experiment that involved Joseph Plateau was a Belgian physicist who did pioneering experimental work on soap bubbles and soap films, which stimulated the mathematical study of bubbles as minimal ![]()
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