Pierre-Louis Lions

Pierre-Louis Lions (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.

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.

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.

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.

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.

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.

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.

The partial differential equation

has received a great deal of attention in the mathematical literature.

Lions' extensive work on this equation is concerned with the existence of rotationally symmetric solutions as well as estimates and existence for boundary value problems of various type.

In the interest of studying solutions on all of Euclidean space, where standard compactness theory does not apply, Lions established a number of compactness results for functions with symmetry.

With Henri Berestycki and Lambertus Peletier, Lions used standard ODE shooting methods to directly study the existence of rotationally symmetric solutions.

However, sharper results were obtained two years later by Berestycki and Lions by variational methods.

They considered the solutions of the equation as rescalings of minima of a constrained optimization problem, based upon a modified Dirichlet energy.

Making use of the Schwarz symmetrization, there exists a minimizing sequence for the infimization problem which consists of positive and rotationally symmetric functions.

So they were able to show that there is a minimum which is also rotationally symmetric and nonnegative.

By adapting the critical point methods of Felix Browder, Paul Rabinowitz, and others, Berestycki and Lions also demonstrated the existence of infinitely many (not always positive) radially symmetric solutions to the PDE.

Maria Esteban and Lions investigated the nonexistence of solutions in a number of unbounded domains with Dirichlet boundary data.

Their basic tool is a Pohozaev-type identity, as previously reworked by Berestycki and Lions.

They showed that such identities can be effectively used with Nachman Aronszajn's unique continuation theorem to obtain the triviality of solutions under some general conditions.

Significant "a priori" estimates for solutions were found by Lions in collaboration with Djairo Guedes de Figueiredo and Roger Nussbaum.

In more general settings, Lions introduced the "concentration-compactness principle", which characterizes when minimizing sequences of functionals may fail to subsequentially converge.

His first work dealt with the case of translation-invariance, with applications to several problems of applied mathematics, including the Choquard equation.

In 1979, Lions married Lila Laurenti, with whom he has one son.

He has also received the French Academy of Science's Prix Paul Doistau–Émile Blutet (in 1986) and Ampère Prize (in 1992).

Lions' parents were Andrée Olivier and the renowned mathematician Jacques-Louis Lions, at the time a professor at the University of Nancy, and from 1991 through 1994 the President of the International Mathematical Union.

He was a recipient of the 1994 Fields Medal and the 1991 Prize of the Philip Morris tobacco and cigarette company.

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 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.

Lions' earliest work dealt with the functional analysis of Hilbert spaces.

Since 2014, he has also been a visiting professor at the University of Chicago.