First he proceeded to a Master's degree and the a doctorate under the supervision of David Gilbarg. He was awarded a Ph.D. in 1966 for his thesis Quasilinear Elliptical Partial Differential Equations in n Variables. Several papers, based on the work of his dissertation, appeared in 1967. First there is the paper On the Dirichlet problem for quasilinear uniformly elliptic equations in n variables in which he extended previous work by his supervisor David Gilbarg, Olga Ladyzhenskaya and others on the solvability of the classical Dirichlet problem in bounded domains for certain second order quasilinear uniformly elliptic equations. Secondly, in the paper The Dirichlet problem for nonuniformly elliptic equation he exploited the maximum principle to formulate general conditions for solvability of the Dirichlet problem for certain nonlinear elliptic equations. In another 1967 paper On Harnack type inequalities and their application to quasilinear elliptic equations Trudinger examines weak solutions, subsolutions and supersolutions of certain quasilinear second order differential equations. In our list of his 1967 papers we mention finally On imbeddings into Orlicz spaces and some applications.
After the award of his doctorate from Stanford University, Trudinger became a Courant Instructor at the Courant Institute of Mathematical Sciences of New York University during the academic year 1966-67. He then returned to Australia where he was appointed as a lecturer at Macquarie University in 1967. He was promoted to Senior Lecturer before moving, in 1970, to the University of Queensland where he was first appointed as a Reader, then promoted to Professor. In 1973 he moved to the Australian National University where he was Head of the Department of Pure Mathematics until 1979. In 1977 Trudinger published an important book in collaboration with David Gilbarg. The book Elliptic partial differential equations of second order aimed to present (in the words of the authors):-
... the systematic development of the general theory of second order quasilinear elliptic equations and of the linear theory required in the process.O John gives an overview of the book as part of his review:-
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.A second edition of this wonderful book appeared in 1983. It had two new chapters one of which examined strong solutions of linear elliptic equations, and the other was on fully nonlinear elliptic equations. A further edition appeared in 1998. In this the authors write:-
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.
This 1998 edition was reprinted in the "Classics in Mathematics" series by Springer-Verlag in 2001.
In following the editions of the famous text by Gilbarg and Trudinger we have become side-tracked from presenting details of Trudinger's career. In 1981 he was honoured by the Australian Mathematical Society when he became the first recipient of their Australian Mathematical Society Medal:-
... 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.In 1982 he became Director of the Centre for Mathematical Analysis at ANU, holding this position until 1990. After a short time away from ANU, he returned as Director of the Centre for Mathematics and its Applications in 1991. In 1992 he became Dean of the School of Mathematical Sciences. Trudinger was elected a fellow of the Australian Academy of Science in 1978 and was awarded their Hannan Medal in 1996. He was also honoured with election as a fellow of the Royal Society of London in 1997. On 24 November 1995 three prizes were awarded by the Institut Henri Poincaré and the publisher Gauthier-Villars, with the support of the Centre National de la Recherche Scientifique. Each prize:-
... 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".Today Trudinger coordinates the Applied and Nonlinear Analysis programme at the Australian National University. We end this biography by quoting the "Highlights" from the web page of the programme:-
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.
Article by: J J O'Connor and E F Robertson