Functions Attached to Geometries

Abstract

During my talk at a conference in New York, NY, in June 1981, when I was presenting my result with FrançoisMURAT on the characterization of effective properties of mixtures of two isotropic conductors, proving and extending the Hashin–Shtrikman bounds, of which I presented the necessary part in Lemma 21.8, David BERGMAN asked me an interesting question, about the meaning of the sequence Aⁿ that I used. I answered him as a joke, that it is the same thing as the thermodynamic limit that he used!¹ Since I think that many do not perceive well what is homogenization and what is not, and it is important notto confuse his approach with homogenization, I shall be more precise. David BERGMAN is a physicist, and what interested hi was a number, for example the current going through two plates with a difference of potential of 1, the zone between theplates being filled with a mixture of two conductors, or the energy stored in the mixture. Although the domain filled with his mixture was fixed and bounded, he invoked a thermodynamic limit, supposed to be the limit of averages on arbitrarily large balls, so I found it a curious idea for a finite domain! However, I guessed that one could make a correct statement by considering a sequence of mixtures using shorter and shorter characteristic lengths, and I considered that his “argument using a thermodynamic limit” meant that he also considered a sequence of mixtures! Of course, what he was doing is not homogenization, since he did not speak of local properties, and of effective coefficients given by symmetric matrices!