BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 44, Number 2, April 2007, Pages 163–232 S 0273-0979(07)01140-8 Article electronically published on January 24, 2007
NONSMOOTH CALCULUS
JUHA HEINONEN
Abstract. We survey recent advances in analysis and geometry, where first order differential analysis has been extended beyond its classical smooth set- tings. Such studies have applications to geometric rigidity questions, but are also of intrinsic interest. The transition from smooth spaces to singular spaces where calculus is possible parallels the classical development from smooth func- tions to functions with weak or generalized derivatives. Moreover, there is a new way of looking at the classical geometric theory of Sobolev functions that is useful in more general contexts.
1. Introduction The word nonsmooth in the title refers both to functions and spaces. Calculus is a field of study where infinitesimal data yields global information. Mathematicians have been practicing calculus with nonsmooth functions for over a century, but only recently in spaces that are not smooth in the traditional sense. In this article, we first survey calculus with nonsmooth functions and then move on to discuss current advances involving singular spaces. Calculus with nonsmooth functions began in the late nineteenth century with attempts to understand the domain of validity for the fundamental theorem of calculus; these studies were not motivated by immediate practical applications. By the 1930s it had become clear that such a calculus was needed in partial differential equations, sciences, and engineering. The so-called direct methods in the calculus of variations argued for the existence of a solution (to a variational problem) which aprioriwas not smooth; the solution was found via a compactness argument in a complete function space. Regularity theory was then developed to show that a posteriori the solution is, under favorable circumstances, infinitely many times differentiable or even real analytic. It was also discovered that in many other cases such smoothness cannot be expected, and the concept of a generalized (or weak, distributional) solution became established in mathematics. The relevant function spaces are the Sobolev spaces, whose development we briefly review in this article.
Received by the editors November 7, 2005, and, in revised form, June 16, 2006. 2000 Mathematics Subject Classification. Primary 28A75, 49J52, 53C23, 51-02. This paper constitutes an expanded version of the AMS invited address given by the author in Boulder, Colorado, in October 2003. The author is grateful for the support and hospitality of MSRI and UC Berkeley, where the bulk of this paper was prepared during a visit in 2002-2003. Supported also by NSF grants DMS 0353549 and DMS 0244421.