![]() ![]() 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. 177-194).Guy David is a teacher and researcher at the Orsay mathematics laboratory (LMO - Université Paris-Saclay, CNRS). Here is a more recent survey by Dierkes: "Singular Minimal Surfaces" (in Geometric Analysis and Nonlinear Partial Differential Equations, Springer (2003), pp. ![]() Jost (ed) Calculus of Variations and Geometric Analysis, Int. A short survey of some old and relatively new results concerning well-posedness of (1)-(3) and its multidimensional analogues can be found in the paper by Dierkes and Huisken, "The N-dimensional analogue of the catenary: Prescribed area", in J. Moreover, the corresponding variational problem has no global solutions for all $A\in\mathbb R$. The equilibrium condition for a hanging heavy surface of constant mass density reads Of the property that an inverted catenary supports smooth rides of a square-wheeledĪ model equation for an inextensible, flexible, heavy surface in a gravitational field was deduced by Poisson Lagrange and later the problem was also studied by Poisson (see the references in the linked papers below). This question arose in imagining a higher-dimensional version I have been unsuccessful in finding anything but simulations of solutions Is there a simple analytic description of any of these surfaces,Īnalogous to the $\cosh$ equation for the catenary curve? Which is the surface of revolution formed by a catenary curve. The middle option above would look something like this when inverted: A square sheet pinned at its four corners.The answer surely depends on how it is pinned to the plane, the boundary Pinned on a plane parallel to the ground, under the influence of gravity? What is the shape taken by an idealized, thin two-dimensional sheet, Is the shape taken by an idealized hanging chain or rope under the influence ![]()
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