Abstract
Haim Brezis who passed away on July 7, 2024, in Jerusalem was one of the world masters of Functional analysis and Nonlinear Partial Differential Equations. He has had an outstanding influence on these fields.
Brezis’s early work focused on the theory of monotone operators. His book, “Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert” (1973), presented the key results that remain useful today. They turned out to provide a fecund framework for optimal transport theory or modeling questions such as pedestrian dynamics. He also worked on various aspects of variational equations and inequalities, establishing optimal regularity results in his doctoral thesis “Problèmes unilatéraux” (1972).
In the 1980s, Brezis tackled elliptic equations invariant under non-compact group actions. Starting with a celebrated work with Louis Nirenberg, Haim Brezis went on to investigate problems arising in surfaces with constant mean curvature (Plateau problem), and harmonic maps in dimension 2. These works involve a deep analysis of critical levels of the energy associated with the problem, proving crucial for overcoming lack of compactness issues. They provide a general method to attack such problems and continue to have a large impact until today.
These works are among the several fundamental contributions of Haim Brezis to the calculus of variations. Another one is the discovery by Haim Brezis and Elliott Lieb in 1983 of an elegant relation between pointwise convergence of functions and convergence of functionals. This result is now classical and has been useful in many applications.
In the 1990s, Haim Brezis studied the Ginzburg-Landau equations related to superconductivity and superfluidity, providing a precise description of the asymptotic behavior of solutions and vortex properties. His book “Ginzburg Landau Vortices” (1994), co-authored with Fabrice Bethuel and Frédéric Hélein, is the key reference for these topics.
A large body of works of Haim Brezis since 1995 has been concerned with fundamental properties of functions spaces for which he established the definitive results. He explored Sobolev spaces of mappings between manifolds, bridging analysis and geometry. It started with a pioneering work with Louis Nirenberg where Haim Brezis gave the minimal regularity for the degree of a map to be defined. He achieved other numerous deep results in this field, including delicate estimates on the Jacobian of a function between spheres with Hoai-Minh Nguyen and, with Luigi Ambrosio and Alessio Figalli, a striking characterization of the perimeter of sets. In a series of works with Jean Bourgain and Petru Mironescu, Haim Brezis derived optimal results regarding the classification of functions spaces and singular behavior of functionals. His work in this area has since found unexpected applications in data analysis and computer science.
Haim Brezis was dedicated to education and research dissemination. His Functional Analysis course in Paris was immensely popular, and his book “Analyse Fonctionnelle – Théorie et Applications” (1983) is a seminal text, translated into numerous languages, and an expanded English version has appeared under the title: “Functional Analysis, Sobolev Spaces and PDEs” (2010). He founded and led the “Progress in Nonlinear Differential Equations and Their Applications” book series and served as editor-in-chief of the “Journal of the European Mathematical Society (JEMS)” from 2003 to 2015, elevating it to a premier publication. He has also served on a large number of editorial boards of scientific journals including Asymptotic Analysis.
Haim Brezis supervised 58 theses, creating a vast academic lineage of over a thousand descendants, in a variety of countries in the word. His achievements were recognized by numerous awards and honorary degrees from eleven universities. He was a member of nine academies, among which the French Academy of Sciences, the European Academy, the National Academy of Sciences, and the American Academy of Arts and Sciences. In January 2024, he was awarded the prestigious Steele Prize for Lifetime Achievement from the American Mathematical Society.
One of the most influential mathematicians of the past fifty years, Haim Brezis leaves behind an extraordinary scientific legacy that will continue to inspire generations of researchers worldwide.
Henri Berestycki and Jean-Michel Coron