3. Overview of the course 13 last part of Chapter 4 touches on a very, very tricky subject, namely the regularity theory for fully nonlinear elliptic equations but, precisely for this reason, the subject is reduced to just a review of the results, and this chapter should not be so difficult to go through. Finally, Chapters 6, 8 and 9 are more advanced especially the last two chapters presuppose some basic notions in partial differential equations and functional analysis, but not more than can be found in any elementary textbook on the subject, like the excellent reference [125], which we shall quote many times. As a general prerequisite for the course, the reader will be assumed to have some basic background knowledge in analysis, especially measure theory, and a little bit of functional analysis. The necessary material can be found in classical pedagogical references such as Rudin [217], Lieb and Loss [178] or Br´ ezis [64] but many other sources would do. On some rare occasions the reader may wish to consult slightly more advanced textbooks such as Rudin [218] (for functional analysis), Evans and Gariepy [128] (for fine regularity properties of convex functions, in particular), Billingsley [41] or Dudley [119] (for probability theory in abstract Polish spaces) but these sources should definitely not be needed in a first reading. If the reader is not familiar with the notion of Hausdorff dimension, we recommend that he/she just forget about it and skip those statements which involve this concept, since they are not of primary importance here. For instance, if a theorem assumes that some measure does not give mass to small sets, just make the stronger assumption that it is absolutely continuous with respect to the Lebesgue measure. There are exercises disseminated all over the text, most of them easy ones whose primary goal is to help the reader’s understanding by making him/her manipulate the basic concepts a little bit. Some exercises however are more tricky. Chapter 10 gathers longer problems, most of which are taken from recent research papers. We chose to gather all these problems together for two reasons: first, because this enables to introduce a great variety of illustrations and applications without being led to an excess of digressions within the main text and secondly, because the solution of one of these problems often requires material which appears at various places in the book. These problems should therefore be of interest for readers who wish to have a synthetic view of the topic. Apart from the index, a “Table of Short Statements” has been added in the back of the book it contains short statements for all the theorems, propositions, definitions, corollaries, etc. which are proven or explained in the main text. Glancing through this table might be a quick way to locate some particular theorem in the book.
Previous Page Next Page