INTRODUCTION 3 The central measure-theoretic techniques for analyzing concentrations and oscillations are set forth in Chapters 4 and 5. I regard these as comprising the core of the current general theory of weak convergence and nonlinear PDE's, augmented in Chapters 2, 3, and 6 by additional techniques available for PDE's with the extra structure of variational and/or maximum principles. Of the several continuing and unifying themes in the exposition two are worth explicit mention here. First, the reader should take note throughout of the primary role of measure theory and real analysis, as opposed to functional analysis. The detailed study of concentration and oscillation phenomena in Chapters 1, 4, and 5 cannot in any obvious way be cast into functional analytic terms. Secondly, note the continual use in quite diverse settings of low amplitude, high frequency periodic test functions to extract useful Information. I have tried to identify throughout the absolutely simplest problems which illustrate various key techniques, and accordingly present in most cases what amount almost to caricatures. I make in particular the Standing assumption that, unless otherwise stated, all given functions are smooth. The notation is mostly either self-explanatory or eise Standard (as in, for instance, Gilbarg- Trudinger [68].) I systematically employ the summation Convention and use the letter C to denote various constants. Although these notes are fairly wide ranging, they by no means exhaust our subject. Let me note in particular my total Omission of relevant Prob- lems in geometry (cf. Aubin [6, 7], Freed-Uhlenbeck [60], Lawson [81], Sacks-Uhlenbeck [105], Taubes [123], etc.), Ã-, G~ and //-convergence (At- touch [5], Murat [96], Spagnolo [113], etc.), relaxation of ill-posed problems (Ball-James [15], Chipot-Kinderlehrer [34], Dacorogna [37], Fonseca [59], Kinderlehrer [75], Kohn-Strang [77], etc.), modification of the Palais-Smale condition (Bahri-Coron [9], [10], Struwe [116], etc.), and new work on the Boltzmann equation (DiPerna-Lions [43, 44]). Various additional references are to be found in the Notes at the end.
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