... the systematic development of the general theory of second order quasilinear elliptic equations and of the linear theory required in the process.

The book is divided into two parts. The first ... is devoted to the linear theory, the second ... to the theory of quasilinear partial differential equations. These 14 chapters are preceded by an Introduction ... which expounds the main ideas and can serve as a guide to the book. The authors restrict themselves mainly to the theory of the Dirichlet problem. With the exception of the prerequisites of basic real analysis and linear algebra the material of this book is almost entirely self-contained. Almost every chapter is concluded by "Notes" (historical and bibliographical remarks, further results) and "Problems". The authors have succeeded admirably in their aims; the book is a real pleasure to read.

The theory of nonlinear elliptic second order equations has continued to flourish during the last fifteen years and, in a brief epilogue to this volume, we signal some of the major advances. Although a proper treatment would necessitate at least another monograph, it is our hope that this book, most of whose text is now more than twenty years old, can continue to serve as background for these and future developments.

Since our first edition we have become indebted to numerous colleagues, all over the globe. It was particularly pleasant in recent years to make and renew friendships with our Russian colleagues, Olga Ladyzhenskaya, ... who have contributed so much to this area. Sadly, we mourn the passing away in 1996 of Ennio De Giorgi, whose brilliant discovery forty years ago opened the door to higher-dimensional nonlinear theory.

... awarded to a member of the Society under the age of 40 years for distinguished research in the mathematical sciences. A significant portion of the research work should have been carried out in Australia.

... carries an award of 10 000 FF, [and] recognizes outstanding articles appearing in each of the three sections of the journal Annales de l'Institut Henri Poincaré. In the nonlinear analysis section, the prize goes to N S Trudinger of the Australian National University for the paper "Isoperimetric inequalities for quermassintegrals".

In recent years, members of the programme have solved major open problems in curvature flow, affine geometry and optimal transportation, using techniques from nonlinear partial differential equations. The first complete proof, for more than two dimensions, of the famous 200 year old Monge problem of mass transfer was found by programme members in 2001.