Pierre-Louis Lions (French: [ljɔ̃ːs]; born 11 August 1956) is a French mathematician. He is known for a number of contributions to the fields of partial differential equations and the calculus of variations. He was a recipient of the 1994 Fields Medal.
Biography Lions entered the École normale supérieure in 1975, and received his doctorate from the University of Pierre and Marie Curie in 1979. He holds the position of Professor of Partial differential equations and their applications at the Collège de France in Paris as well as a position at École Polytechnique. Since 2014, he has also been a visiting professor at the University of Chicago. In 1979, Lions married Lila Laurenti, with whom he has one son. Lions' parents were Andrée Olivier and the renowned mathematician Jacques-Louis Lions, at the time a professor at the University of Nancy.
Awards and honors In 1994, while working at the Paris Dauphine University, Lions received the International Mathematical Union's prestigious Fields Medal. He was cited for his contributions to viscosity solutions, the Boltzmann equation, and the calculus of variations. He has also received the French Academy of Sciences' Prix Paul Doistau–Émile Blutet (in 1986) and Ampère Prize (in 1992). He was an invited professor at the Conservatoire national des arts et métiers (2000). He is a doctor honoris causa of Heriot-Watt University (Edinburgh), EPFL (2010), Narvik University College (2014), and of the City University of Hong-Kong and is listed as an ISI highly cited researcher.
Mathematical work
Operator theory Lions' earliest work dealt with the functional analysis of Hilbert spaces. His first published article, in 1977, was a contribution to the vast literature on convergence of certain iterative algorithms to fixed points of a given nonexpansive self-map of a closed convex subset of Hilbert space.[L77] In collaboration with his thesis advisor Haïm Brézis, Lions gave new results about maximal monotone operators in Hilbert space, proving one of the first convergence results for Bernard Martinet and R. Tyrrell Rockafellar's proximal point algorithm.[BL78] In the time since, there have been a large number of modifications and improvements of such results. With Bertrand Mercier, Lions proposed a "forward-backward splitting algorithm" for finding a zero of the sum of two maximal monotone operators.[LM79] Their algorithm can be viewed as an abstract version of the well-known Douglas−Rachford and Peaceman−Rachford numerical algorithms for computation of solutions to parabolic partial differential equations. The Lions−Mercier algorithms and their proof of convergence have been particularly influential in the literature on operator theory and its applications to numerical analysis. A similar method was studied at the same time by Gregory Passty.
Calculus of variations The mathematical study of the steady-state Schrödinger–Newton equation, also called the Choquard equation, was initiated in a seminal article of Elliott Lieb. It is inspired by plasma physics via a standard approximation technique in quantum chemistry. Lions showed that one could apply standard methods such as the mountain pass theorem, together with some technical work of Walter Strauss, in order to show that a generalized steady-state Schrödinger–Newton equation with a radially symmetric generalization of the gravitational potential is necessarily solvable by a radially symmetric function.[L80] The partial differential equation
∂ 2 u ∂ x 1 2 + ⋯ + ∂ 2 u ∂ x n 2 = f ( u ) {\displaystyle {\frac {\partial ^{2}u}{\partial x_{1}^{2}}}+\cdots +{\frac {\partial ^{2}u}{\partial x_{n}^{2}}}=f(u)}
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