Emil Artin had been forced to leave Germany in 1937 because of Hitler's anti-Semitic legislation and, after teaching at Notre Dame for the academic year 1937-38, he arrived in Bloomington in 1938. Artin became Gilbarg's thesis advisor and, after undertaking research in algebra, Gilbarg was awarded his doctorate in 1941 for his thesis on algebraic number theory On the structure of the group of p-adic 1-units. He was appointed to the faculty at Indiana University, but soon political events would create a major change in his career.
Had it not been for World War II, Gilbarg would almost certainly be known today as an algebraist. The United States, however, following the Japanese attack on the American fleet at Pearl Harbour in December 1941, entered World War II and in 1942 Gilbarg began undertaking war work with the Bureau of Standards and then at the Naval Ordnance Laboratory. At the Naval Ordnance Laboratory he became head of the fluid dynamics and theoretical mechanics section. His was work there took him into new areas of mathematics and involved fluid dynamics and nonlinear partial differential equations. Except for a paper relating to his thesis which was published in the Duke Mathematical Journal in 1942, all his remaining mathematical publications were in the areas of fluid dynamics and nonlinear partial differential equations.
After World War II ended, Gilbarg returned to his position as an assistant professor at Indiana University. His first paper on fluid dynamics was published in 1947, entitled On the flow patterns common to certain classes of plane fluid motions. A J McConnell writes in a review:-
It is proved that (i) the stream functions of any two steady plane liquid flows that possess the same streamlines must be connected by a linear relation, except for the special case of those flows having constant velocity along the streamlines; (ii) the only flow patterns common to the potential flows of gases and liquids are those corresponding to circular vortex and radial flows.In 1949 Gilbarg published A generalization of the Schwarz-Christoffel transformation, and A characterization of non-isentropic irrotational flows. Clifford Truesdell begins a review of the second of these papers by noting:-
This paper is unusual in that it attacks a problem of gas dynamics in the large.Although continuing to hold his position on the faculty at Indiana University, Gilbarg became more and more involved with working with mathematicians at Stanford University :-
After several summer visits to Stanford, and one full-year visit in the 1954-55 Academic Year, there was a firmly established mutual regard between Gilbarg and the Stanford Mathematics faculty - at that time including such outstanding figures as George Pólya, Gabor Szegő, Charles Loewner, Stefan Bergman and Max Schiffer - and in 1957 he accepted an invitation to take up an appointment as Professor of Mathematics, a position which he held until becoming Professor Emeritus of Mathematics in 1989.For many mathematicians, Gilbarg is best known for his remarkable book Elliptic Partial Differential Equations of Second Order written in collaboration with Neil Trudinger and published in 1977. Trudinger had written his Ph.D. thesis at Stanford University advised by Gilbarg. He was awarded his doctorate in 1966 and, after working at a number of different institutions, returned to Stanford where he taught between 1971 and 1973. Trudinger received the American Mathematical Society's Steele Prize for Mathematical Exposition in 2008 for their jointly authored book. Sadly Gilbarg could not share this award as he died seven years earlier. In  Trudinger explained how the book came to be written:-
I could never have imagined forty years ago when my book with David Gilbarg on elliptic partial differential equations was first published that it would get such recognition. The book was originally conceived by us after I had prepared lecture notes for the spring quarter of the graduate PDE course at Stanford in 1971. My topics were Sobolev spaces and their application to linear elliptic PDE, and we decided to start by blending these with earlier notes of Dave Gilbarg on the Schauder theory. Six years later and after a lot of hard work, including long and painful negotiations over language, the first edition appeared.In this first edition the authors explained that their aim was to present:-
... the systematic development of the general theory of second order quasilinear elliptic equations and of the linear theory required in the process.In 1983 a second, revised edition, was published containing additional material. A reviewer writes of this second edition:-
Primarily addressed to graduate students this elegant book is accessible and useful to a broad spectrum of applied mathematicians.The authors of  write:-
Aside from his fluid dynamics work, undoubtedly [Gilbarg's] best-known contribution to the mathematical literature is the monograph "Elliptic Partial Differential Equations of Second Order," co-authored with his former Stanford PhD student Neil S Trudinger. Since 1977 when it first appeared, this book has seen several reprintings, and is one of the most frequently cited graduate-level texts ever published in the mathematical literature. At the recent commemorative conference in the Stanford Mathematics Department, James Serrin described the Gilbarg-Trudinger text as being "on the bookshelf of everyone working in partial differential equations, a monumental work which is one of the great lasting achievements of analysis."In  there is a quotation from Leon Simon, who was then chairman of the Stanford Department of Mathematics, concerning the influence that Gilbarg had on bringing outstanding mathematicians to Stanford:-
He brought to Stanford some of the biggest names in the country, even internationally. It was an important period that helped make Stanford one of the leading mathematics departments in the nation.During an eleven year period when Gilbarg was Chairman of the Stanford Department of Mathematics, namely 1959-1970, appointments were made to the Department including Paul Cohen, Lars Hörmander, Kunichiro Kodaira, Donald Ornstein, Ralph Phillips, Hans Samelson, and Donald Spencer. Hans Samelson said :-
He was a very good chairman, well organized, with good judgment, always looking out for the members of the department, with high standards for research and teaching. When I came here, he was very helpful and made me feel at home right away. He had a wonderful memory for what had happened at department [or other] meetings, how and when and by whom decisions were made.Gilbarg was married to Shirley; they had one son Daniel who is a professor of sociology.
The authors of  end their Memorial Resolution as follows:-
Apart from his central role in the development and support of the Mathematics Department at Stanford, David Gilbarg was much admired by friends and colleagues for his personal qualities, including his selfless dedication to the well-being of the department, his love of mathematics generally and his warm support of the work of his colleagues. These qualities were a recurring theme in the many tributes from family, friends and colleagues who attended a recent commemorative event hosted by the Mathematics Department. Particularly warm tributes were paid by former PhD students, citing Gilbarg's ability to convey a love of mathematics, his unstinting support, his encouragement in development of not only their general mathematical skills but also the ability to recognize and use the important key ideas within a field, and his insistence on exactness and clarity in their writing.
Article by: J J O'Connor and E F Robertson
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