On Haim Brezis A volume offering a perspective on the state of the art of nonlinear partial differential equations is a fitting tribute to someone whose ideas and contributions are so ubiquitous throughout the subject. All the more so, it seemed to us, when these perspectives aim at shedding light on future directions and challenges that lie ahead of the subject. Ha1m Brezis is indeed known to have relentlessly scrutinized the mathematical horizon and has always shown an unquenchable passion for open problems. To describe in any kind of detail how much the field of nonlinear PDE's owes to Ha1m Brezis would be an immense task, one that goes well beyond the scope of this brief introduction. Nor is it our purpose here. We only wish to paint with a few broad brush strokes some of the landscape that Ha1m's activity has profoundly changed (and sometimes created), leaving the finer detail for others to fill in. His list of publications - already incomplete! - that is included at the end of this volume is the prime source for more detailed information. A striking feature of Ha!m's many contributions is the variety of problems to which he has made seminal contributions. The spectrum of his work spans from the very abstract to specific equations arising in physics. Ha1m grew up as a mathemati- cian in the effervescent environment created by Laurent Schwartz, Gustave Choquet and Jacques-Louis Lions in Paris. He started his career as a junior researcher at the French Centre National de la Recherche Scientifique. As Jacques-Louis Lions had strong ties with Guido Stampacchia and with the Italian school in general, Ha1m Brezis had the occasion early on to travel to Pisa. Under their joint influence, he became interested in problems from mechanics, involving unilateral constraints and variational inequalities. In collaboration with Guido Stampacchia and later with David Kinderleher, in particular, he established regularity results for variational inequalities involving the Dirichlet integral that proved to be important for deriv- ing estimates for the regularity of free boundaries. Continuing in this direction, he developed applications of variational inequalities to specific problems in mechanics such as subsonic flow around a profile (using the hodograph method), flow of water through a porous dam and the evolution of ice-water mixtures. Soon afterwards, Ha!m achieved his first major results in the theory of maxi- mal monotone operators and their generalizations and in the theory of semigroups of contractions. These bear the strong influence of Felix Browder. This program aimed at laying the theoretical foundations of evolution equations involving opera- tors of monotone type. It led to a unified framework for a wide range of nonlinear ix
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