View source for Pierre-Louis Lions

{{short description|French mathematician (born 1956)}}
{{Infobox scientist
| name              = Pierre-Louis Lions
| image             = File:Pierre-Louis Lions par Philippe Binant.jpg
| caption           = Lions in 2005
| birth_date        = {{Birth date and age|df=y|1956|08|11}}
| birth_place       = [[Grasse]], [[Alpes-Maritimes]], [[French Fourth Republic|France]]
| death_date        = 
| death_place       = 
| residence         = 
| citizenship       = 
| ethnicity         = 
| field             = [[Mathematics]]
| work_institutions = [[Collège de France]]<br />[[École Polytechnique]]<br />[[Université Paris-Dauphine|University of Paris-Dauphine]]
| education         = [[Lycée Louis-le-Grand]]
| alma_mater        = [[École normale supérieure (Paris)|École normale supérieure]]<br>[[Pierre and Marie Curie University]]
| doctoral_advisor  = [[Haïm Brezis]]
| doctoral_students = [[María J. Esteban]]<br />[[Olivier Guéant]]<br />[[Gilles Motet]]<br />[[Benoit Perthame]]<br />[[Nader Masmoudi]]<br />[[Cédric Villani]]
| known_for         = [[Nonlinear partial differential equation]]s<br />[[Mean field game theory]]<br />[[Viscosity solution]]
| prizes            = [[List of International Congresses of Mathematicians Plenary and Invited Speakers|ICM Speaker]] (1983, 1990, 1994)<br>[[Peccot Lectures|Peccot Lecture]] (1983)<br>{{no wrap|[[Prix Paul Doistau–Émile Blutet]] (1986)}}<br>[[Ampère Prize]] (1992)<br>[[Fields Medal]] (1994)
| thesis_title      = Sur quelques classes d'équations aux dérivées partielles non linéaires et leur résolution numérique
| thesis_year       = 1979
}}
'''Pierre-Louis Lions''' ({{IPA|fr|ljɔ̃ːs|lang}};<ref>[https://www.youtube.com/watch?v=czl1a084HGQ CORE Fields Medal Talk: Pierre-Louis Lions on Mean Field Games]<!--at 0:13--></ref> born 11 August 1956) is a French [[mathematician]]. He is known for a number of contributions to the fields of [[partial differential equation]]s and the [[calculus of variations]]. He was a recipient of the 1994 [[Fields Medal]] and the 1991 Prize of the [[Altria|Philip Morris]] tobacco and cigarette company.<ref>{{Cite web|url=https://www.ae-info.org/ae/Member/Lions_Pierre-Louis|title = Academy of Europe: Lions Pierre-Louis}}</ref>

==Biography==
Lions entered the [[École normale supérieure (Paris)|École normale supérieure]] in 1975, and received his doctorate from the [[University of Pierre and Marie Curie]] in 1979.<ref>{{Cite web |url=http://www2.cnrs.fr/sites/communique/fichier/medailles_fields_francaises_corrige.pdf |title=La Médaille Fields : 11 lauréats sur 44 sont issus de laboratoires français., Alain Connes |work=www2.cnrs.fr  | accessdate=11 May 2010}}</ref> 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]].<ref>{{cite web |title=Pierre-Louis Lions - Biographie |url=https://www.college-de-france.fr/site/pierre-louis-lions/biographie.htm |publisher=Collège de France |access-date=16 November 2020 |language=fr}}</ref><ref name=":0">{{cite web |title=Pierre-Louis Lions |url=https://stevanovichcenter.uchicago.edu/pierre-louis-lions/ |publisher=University of Chicago |access-date=16 November 2020}}</ref> Since 2014, he has also been a visiting professor at the [[University of Chicago]].<ref>{{cite web |title=Fields Medal |url=https://www.uchicago.edu/about/accolades/14/ |publisher=University of Chicago |access-date=16 November 2020}}</ref>

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 solution]]s, the [[Boltzmann equation]], and the [[calculus of variations]]. He has also received the [[French Academy of Science]]'s [[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).<ref>Pierre-Louis Lions, « Analyse, modèles et simulations », ''Université de tous les savoirs'', '''4''', 86-92, Éditions Odile Jacob, Paris, 2001.</ref> He is a doctor honoris causa of [[Heriot-Watt University]]<ref>{{Cite web|url=http://www.ae-info.org/ae/User/Lions_Pierre-Louis|title=Academy of Europe: Lions Pierre-Louis|last=Hoffmann|first=Ilire Hasani, Robert|website=www.ae-info.org|access-date=2016-04-06}}</ref> ([[Edinburgh]]), [[École Polytechnique Fédérale de Lausanne|EPFL]] (2010),<ref>{{Cite web|last=Pousaz|first=Lionel|date=2010-11-10|title=The "Magistrale" crowns the founder of Yahoo|url=https://actu.epfl.ch/news/the-magistrale-crowns-the-founder-of-yahoo/|language=en}}</ref> [[Narvik University College]] (2014), and of the [[City University of Hong Kong|City University of Hong-Kong]] and is listed as an [[ISI highly cited researcher]].<ref>{{Citation
 |last        = Thomson ISI
 |title       = Lions, Pierre-Louis, ISI Highly Cited Researchers
 |url         = http://hcr3.isiknowledge.com/author.cgi?&link1=Browse&link2=Results&id=3423
 |access-date  = 2009-06-20
 |url-status     = dead
 |archive-url  = https://web.archive.org/web/20060304074201/http://hcr3.isiknowledge.com/author.cgi?&link1=Browse&link2=Results&id=3423
 |archive-date = 2006-03-04
}}
</ref>

==Mathematical work==
===Operator theory===
Lions' earliest work dealt with the [[functional analysis]] of [[Hilbert space]]s. His first published article, in 1977, was a contribution to the vast literature on convergence of certain iterative algorithms to [[Fixed point (mathematics)|fixed point]]s of a given [[Metric map|nonexpansive self-map]] of a closed convex subset of Hilbert space.{{ran|L77}}<ref>{{cite journal|mr=1911872|last1=Xu|first1=Hong-Kun|title=Iterative algorithms for nonlinear operators|journal=[[Journal of the London Mathematical Society]]|series=Second Series|volume=66|year=2002|issue=1|pages=240–256|doi=10.1112/S0024610702003332|zbl=1013.47032|s2cid=122667025 }}</ref> In collaboration with his thesis advisor [[Haïm Brézis]], Lions gave new results about [[Monotonic_function#Monotonicity_in_functional_analysis|maximal monotone operators]] in Hilbert space, proving one of the first convergence results for Bernard Martinet and [[R. Tyrrell Rockafellar]]'s [[Proximal operator|proximal point algorithm]].{{ran|BL78}}<ref name="eckber">{{cite journal|mr=1168183|last1=Eckstein|first1=Jonathan|last2=Bertsekas|first2=Dimitri P.|title=On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators|journal=[[Mathematical Programming]]|volume=55|year=1992|issue=3|series=Series A|pages=293–318|doi=10.1007/BF01581204|author-link2=Dimitri Bertsekas|citeseerx=10.1.1.85.9701|zbl=0765.90073|s2cid=15551627 }}</ref> In the time since, there have been a large number of modifications and improvements of such results.<ref>{{cite journal|mr=1734665|last1=Solodov|first1=M. V.|last2=Svaiter|first2=B. F.|title=Forcing strong convergence of proximal point iterations in a Hilbert space|journal=[[Mathematical Programming]]|volume=87|year=2000|issue=1|series=Series A|pages=189–202|doi=10.1007/s101079900113|zbl=0387.47038|s2cid=106476 }}</ref>

With Bertrand Mercier, Lions proposed a "forward-backward splitting algorithm" for finding a zero of the sum of two maximal monotone operators.{{ran|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 equation]]s. 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.<ref>{{cite journal|first1=Gregory B.|last1=Passty|title=Ergodic convergence to a zero of the sum of monotone operators in Hilbert space|journal=[[Journal of Mathematical Analysis and Applications]]|volume=72|year=1979|issue=2|pages=383–390|zbl=0428.47039|doi=10.1016/0022-247X(79)90234-8|doi-access=free|mr=0559375}}</ref><ref name="eckber" />

===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 H. Lieb|Elliott Lieb]].<ref>Elliott H. Lieb. Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation. Studies in Appl. Math. 57 (1976/77), no. 2, 93–105.</ref> It is inspired by [[plasma physics]] via a [[Hartree–Fock method|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 Alexander Strauss|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.{{ran|L80}}

The partial differential equation
:<math>\frac{\partial^2u}{\partial x_1^2}+\cdots+\frac{\partial^2u}{\partial x_n^2}=f(u)</math>
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.{{ran|L82a}} 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.{{ran|L82b}} With [[Henri Berestycki]] and [[:de:Lambertus Peletier|Lambertus Peletier]], Lions used standard ODE [[shooting method]]s to directly study the existence of rotationally symmetric solutions.{{ran|BLP81}} 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.{{ran|BL83a}} 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.{{ran|BL83b}} [[María J. Esteban|Maria Esteban]] and Lions investigated the nonexistence of solutions in a number of unbounded domains with Dirichlet boundary data.{{ran|EL82}} Their basic tool is a Pohozaev-type identity, as previously reworked by Berestycki and Lions.{{ran|BL83a}} 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.<ref>N. Aronszajn. A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order. J. Math. Pures Appl. (9) 36 (1957), 235–249.</ref> Significant "a priori" estimates for solutions were found by Lions in collaboration with [[Djairo Guedes de Figueiredo]] and [[Roger D. Nussbaum|Roger Nussbaum]].{{ran|FLN82}}

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.{{ran|L84a}} He was also able to extend parts of his work with Berestycki to settings without any rotational symmetry.{{ran|L84b}} By making use of [[Abbas Bahri]]'s topological methods and min-max theory, Bahri and Lions were able to establish multiplicity results for these problems.{{ran|BL88}} Lions also considered the problem of dilation invariance, with natural applications to optimizing functions for dilation-invariant functional inequalities such as the [[Sobolev inequality]].{{ran|L85a}} He was able to apply his methods to give a new perspective on previous works on geometric problems such as the [[Yamabe problem]] and [[harmonic map]]s.{{ran|L85b}} With Thierry Cazenave, Lions applied his concentration-compactness results to establish [[orbital stability]] of certain symmetric solutions of [[nonlinear Schrödinger equation]]s which admit variational interpretations and energy-conserving solutions.{{ran|CL82}}

===Transport and Boltzmann equations===
In 1988, [[François Golse]], Lions, [[Benoit Perthame|Benoît Perthame]], and Rémi Sentis studied the [[transport equation]], which is a first-order linear partial differential equation.{{ran|GLPS88}} They showed that if the first-order coefficients are randomly chosen according to some [[probability distribution]], then the corresponding function values are distributed with regularity which is enhanced from the original probability distribution. These results were later extended by DiPerna, Lions, and Meyer.{{ran|DLM91}} In the physical sense, such results, known as ''velocity-averaging lemmas'', correspond to the fact that macroscopic observables have greater smoothness than their microscopic rules directly indicate. According to [[Cédric Villani]], it is unknown if it is possible to instead use the explicit representation of solutions of the transport equation to derive these properties.<ref name="villani">{{cite encyclopedia|first1=Cédric|last1=Villani|title=A review of mathematical topics in collisional kinetic theory|encyclopedia=Handbook of mathematical fluid dynamics, Vol. I|series=Handbook of Mathematical Fluid Dynamics |pages=71–305|publisher=[[North-Holland]]|location=Amsterdam|year=2002|volume=1 |author-link1=Cedric Villani|mr=1942465|doi=10.1016/S1874-5792(02)80004-0|editor-first1=S.|editor-last1=Friedlander|editor-first2=D.|editor-last2=Serre|isbn=0-444-50330-7|editor-link1=Susan Friedlander|editor-link2=Denis Serre|s2cid=117660436|zbl=1170.82369}}</ref>

The classical [[Picard–Lindelöf theorem]] deals with integral curves of [[Lipschitz continuity|Lipschitz-continuous]] [[vector field]]s. By viewing integral curves as [[Method of characteristics|characteristic curves]] for a transport equation in multiple dimensions, Lions and [[Ronald DiPerna]] initiated the broader study of integral curves of [[Sobolev space|Sobolev]] vector fields.{{ran|DL89a}} DiPerna and Lions' results on the transport equation were later extended by [[Luigi Ambrosio]] to the setting of [[bounded variation]], and by [[Alessio Figalli]] to the context of [[stochastic process]]es.<ref>{{cite book|mr=3443169|last1=Bogachev|first1=Vladimir I.|last2=Krylov|first2=Nicolai V.|author-link2=Nikolay Krylov (mathematician, born 1941)|last3=Röckner|first3=Michael|last4=Shaposhnikov|first4=Stanislav V.|title=Fokker–Planck–Kolmogorov equations|series=[[Mathematical Surveys and Monographs]]|volume=207|publisher=[[American Mathematical Society]]|location=Providence, RI|year=2015|isbn=978-1-4704-2558-6|doi=10.1090/surv/207|author-link3=Michael Röckner|zbl=1342.35002}}</ref>

DiPerna and Lions were able to prove the global existence of solutions to the [[Boltzmann equation]].{{ran|DL89b}} Later, by applying the methods of [[Fourier integral operator]]s, Lions established estimates for the Boltzmann collision operator, thereby finding compactness results for solutions of the Boltzmann equation.{{ran|L94}} As a particular application of his compactness theory, he was able to show that solutions subsequentially converge at infinite time to Maxwell distributions.<ref name="villani" /> DiPerna and Lions also established a similar result for the [[Vlasov equation|Maxwell−Vlasov equations]].{{ran|DL89c}}<ref>{{cite book|mr=1379589|last1=Glassey|first1=Robert T.|title=The Cauchy problem in kinetic theory|publisher=[[Society for Industrial and Applied Mathematics]]|location=Philadelphia, PA|year=1996|isbn=0-89871-367-6|doi=10.1137/1.9781611971477|zbl=0858.76001}}</ref>

===Viscosity solutions===
[[Michael G. Crandall|Michael Crandall]] and Lions introduced the notion of [[viscosity solution]], which is a kind of generalized solution of [[Hamilton–Jacobi equation]]s. Their definition is significant since they were able to establish a [[well-posedness]] theory in such a generalized context.{{ran|CL83}} The basic theory of viscosity solutions was further worked out in collaboration with [[Lawrence Evans]].{{ran|CEL84}} Using a min-max quantity, Lions and [[:fr:Jean-Michel Lasry|Jean-Michel Lasry]] considered mollification of functions on [[Hilbert space]] which preserve analytic phenomena.{{ran|LL86}} Their approximations are naturally applicable to Hamilton-Jacobi equations, by regularizing sub- or super-solutions. Using such techniques, Crandall and Lions extended their analysis of Hamilton-Jacobi equations to the infinite-dimensional case, proving a comparison principle and a corresponding uniqueness theorem.{{ran|CL85}}

Crandall and Lions investigated the numerical analysis of their viscosity solutions, proving convergence results both for a [[finite difference]] scheme and [[Numerical diffusion|artificial viscosity]].{{ran|CL84}}

The comparison principle underlying Crandall and Lions' notion of viscosity solution makes their definition naturally applicable to second-order [[elliptic partial differential equation]]s, given the [[maximum principle]].<ref>Hitoshi Ishii. On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs. Comm. Pure Appl. Math. 42 (1989), no. 1, 15–45.</ref>{{ran|IL90}} Crandall, Ishii, and Lions' survey article on viscosity solutions for such equations has become a standard reference work.{{ran|CIL92}}

===Mean field games===
With Jean-Michel Lasry, Lions has contributed to the development of [[mean-field game theory]].{{ran|LL07}}

==Major publications==
{{refbegin}}
'''Articles.'''
{{refbegin|30em}}
{{longitem|{{rma|L77|tw=4.5em|Pierre-Louis Lions. ''[https://gallica.bnf.fr/ark:/12148/bpt6k5731057m/f55.item Approximation de points fixes de contractions.]'' C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 21, A1357–A1359.}}}}
{{longitem|{{rma|BL78|tw=4.5em|H. Brézis and P.L. Lions. ''Produits infinis de résolvantes.'' [[Israel Journal of Mathematics|Israel J. Math.]] 29 (1978), no. 4, 329–345. [[doi:10.1007/BF02761171]]}}}}
{{longitem|{{rma|LM79|tw=4.5em|P.L. Lions and B. Mercier. ''Splitting algorithms for the sum of two nonlinear operators.'' SIAM J. Numer. Anal. 16 (1979), no. 6, 964–979. [[doi:10.1137/0716071]]}}}}
{{longitem|{{rma|L80|tw=4.5em|P.L. Lions. ''The Choquard equation and related questions.'' Nonlinear Anal. 4 (1980), no. 6, 1063–1072. [[doi:10.1016/0362-546X(80)90016-4]]}}}}
{{longitem|{{rma|BLP81|tw=4.5em|H. Berestycki, P.L. Lions, and L.A. Peletier. ''An ODE approach to the existence of positive solutions for semilinear problems in {{math|'''R'''<sup>N</sup>}}.'' Indiana Univ. Math. J. 30 (1981), no. 1, 141–157. [[doi:10.1512/iumj.1981.30.30012]]}}}}
{{longitem|{{rma|CL82|tw=4.5em|T. Cazenave and P.-L. Lions. ''Orbital stability of standing waves for some nonlinear Schrödinger equations.'' Comm. Math. Phys. 85 (1982), no. 4, 549–561. [[doi:10.1007/bf01403504]]}}}}
{{longitem|{{rma|EL82|tw=4.5em|M.J. Esteban and P.L. Lions. ''Existence and nonexistence results for semilinear elliptic problems in unbounded domains.'' Proc. Roy. Soc. Edinburgh Sect. A 93 (1982), no. 1-2, 1–14. [[doi:10.1017/S0308210500031607]]}}}}
{{longitem|{{rma|FLN82|tw=4.5em|D.G. de Figueiredo, P.-L. Lions, and R.D. Nussbaum. ''A priori estimates and existence of positive solutions of semilinear elliptic equations.'' J. Math. Pures Appl. (9) 61 (1982), no. 1, 41–63. [[doi:10.1007/978-3-319-02856-9_11]]}}}}
{{longitem|{{rma|L82a|tw=4.5em|P.L. Lions. ''On the existence of positive solutions of semilinear elliptic equations.'' SIAM Rev. 24 (1982), no. 4, 441–467. [[doi:10.1137/1024101]]}}}}
{{longitem|{{rma|L82b|tw=4.5em|Pierre-Louis Lions. ''Symétrie et compacité dans les espaces de Sobolev.'' J. Functional Analysis 49 (1982), no. 3, 315–334. [[doi:10.1016/0022-1236(82)90072-6]]}}}}
{{longitem|{{rma|BL83a|tw=4.5em|H. Berestycki and P.-L. Lions. ''Nonlinear scalar field equations. I.'' Arch. Rational Mech. Anal. 82 (1983), no. 4, 313–345. [[doi:10.1007/BF00250555]]}}}}
{{longitem|{{rma|BL83b|tw=4.5em|H. Berestycki and P.-L. Lions. ''Nonlinear scalar field equations. II.'' Arch. Rational Mech. Anal. 82 (1983), no. 4, 347–375. [[doi:10.1007/BF00250556]]}}}}
{{longitem|{{rma|CL83|tw=4.5em|Michael G. Crandall and Pierre-Louis Lions. ''Viscosity solutions of Hamilton-Jacobi equations.'' Trans. Amer. Math. Soc. 277 (1983), no. 1, 1–42. [[doi:10.1090/S0002-9947-1983-0690039-8]]}}}}
{{longitem|{{rma|CEL84|tw=4.5em|M.G. Crandall, L.C. Evans, and P.-L. Lions. ''Some properties of viscosity solutions of Hamilton-Jacobi equations.'' Trans. Amer. Math. Soc. 282 (1984), no. 2, 487–502. [[doi:10.1090/S0002-9947-1984-0732102-X]]}}}}
{{longitem|{{rma|CL84|tw=4.5em|M.G. Crandall and P.-L. Lions. ''Two approximations of solutions of Hamilton-Jacobi equations.'' Math. Comp. 43 (1984), no. 167, 1–19. [[doi:10.1090/S0025-5718-1984-0744921-8]]}}}}
{{longitem|{{rma|L84a|tw=4.5em|P.-L. Lions. ''The concentration-compactness principle in the calculus of variations. The locally compact case. I.'' Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109–145. [[doi:10.1016/S0294-1449(16)30428-0]]}}}}
{{longitem|{{rma|L84b|tw=4.5em|P.-L. Lions. ''The concentration-compactness principle in the calculus of variations. The locally compact case. II.'' Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223–283. [[doi:10.1016/S0294-1449(16)30422-X]]}}}}
{{longitem|{{rma|CL85|tw=4.5em|Michael G. Crandall and Pierre-Louis Lions. ''Hamilton-Jacobi equations in infinite dimensions. I. Uniqueness of viscosity solutions.'' J. Funct. Anal. 62 (1985), no. 3, 379–396. [[doi:10.1016/0022-1236(85)90011-4]]}}}}
{{longitem|{{rma|L85a|tw=4.5em|P.-L. Lions. ''The concentration-compactness principle in the calculus of variations. The limit case. I.'' Rev. Mat. Iberoamericana 1 (1985), no. 1, 145–201. [[doi:10.4171/RMI/6]]}}}}
{{longitem|{{rma|L85b|tw=4.5em|P.-L. Lions. ''The concentration-compactness principle in the calculus of variations. The limit case. II.'' Rev. Mat. Iberoamericana 1 (1985), no. 2, 45–121. [[doi:10.4171/RMI/12]]}}}}
{{longitem|{{rma|LL86|tw=4.5em|J.-M. Lasry and P.-L. Lions. ''A remark on regularization in Hilbert spaces.'' [[Israel Journal of Mathematics|Israel J. Math.]] 55 (1986), no. 3, 257–266. [[doi:10.1007/BF02765025]]}}}}
{{longitem|{{rma|BL88|tw=4.5em|A. Bahri and P.-L. Lions. ''Morse index of some min-max critical points. I. Application to multiplicity results.'' Comm. Pure Appl. Math. 41 (1988), no. 8, 1027–1037. [[doi:10.1002/cpa.3160410803]]}}}}
{{longitem|{{rma|GLPS88|tw=4.5em|François Golse, Pierre-Louis Lions, Benoît Perthame, and Rémi Sentis. ''Regularity of the moments of the solution of a transport equation.'' [[Journal of Functional Analysis|J. Funct. Anal.]] 76 (1988), no. 1, 110–125. [[doi:10.1016/0022-1236(88)90051-1]]}}}}
{{longitem|{{rma|ATL89|tw=4.5em|A. Alvino, G. Trombetti, and P.-L. Lions. ''On optimization problems with prescribed rearrangements.'' Nonlinear Anal. 13 (1989), no. 2, 185–220. [[doi:10.1016/0362-546X(89)90043-6]]}}}}
{{longitem|{{rma|DL89a|tw=4.5em|R.J. DiPerna and P.L. Lions. ''Ordinary differential equations, transport theory and Sobolev spaces.'' Invent. Math. 98 (1989), no. 3, 511–547. [[doi:10.1007/BF01393835]]}}}}
{{longitem|{{rma|DL89b|tw=4.5em|R.J. DiPerna and P.-L. Lions. ''On the Cauchy problem for Boltzmann equations: global existence and weak stability.'' Ann. of Math. (2) 130 (1989), no. 2, 321–366. [[doi:10.2307/1971423]]}}}}
{{longitem|{{rma|DL89c|tw=4.5em|R.J. DiPerna and P.-L. Lions. ''Global weak solutions of Vlasov-Maxwell systems.'' Comm. Pure Appl. Math. 42 (1989), no. 6, 729–757. [[doi:10.1002/cpa.3160420603]]}}}}
{{longitem|{{rma|ATL90|tw=4.5em|A. Alvino, G. Trombetti, and P.-L. Lions. ''Comparison results for elliptic and parabolic equations via Schwarz symmetrization.'' Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), no. 2, 37–65. [[doi:10.1016/S0294-1449(16)30303-1]]}}}}
{{longitem|{{rma|IL90|tw=4.5em|H. Ishii and P.-L. Lions. ''Viscosity solutions of fully nonlinear second-order elliptic partial differential equations.'' J. Differential Equations 83 (1990), no. 1, 26–78. [[doi:10.1016/0022-0396(90)90068-Z]]}}}}
{{longitem|{{rma|DLM91|tw=4.5em|R.J. DiPerna, P.L. Lions, and Y. Meyer. ''{{math|L<sup>p</sup>}} regularity of velocity averages.'' Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), no. 3-4, 271–287. [[doi:10.1016/s0294-1449(16)30264-5]]}}}}
{{longitem|{{rma|CIL92|tw=4.5em|Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions. ''User's guide to viscosity solutions of second order partial differential equations.'' Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. [[doi:10.1090/S0273-0979-1992-00266-5]]}}}}
{{longitem|{{rma|L94|tw=4.5em|P.-L. Lions. ''Compactness in Boltzmann's equation via Fourier integral operators and applications. I.'' J. Math. Kyoto Univ. 34 (1994), no. 2, 391–427. [[doi:10.1215/kjm/1250519017]]}}}}
{{longitem|{{rma|LL06a|tw=4.5em|Jean-Michel Lasry and Pierre-Louis Lions. ''Jeux à champ moyen. I. Le cas stationnaire.'' C. R. Math. Acad. Sci. Paris 343 (2006), no. 9, 619–625. [[doi:10.1016/j.crma.2006.09.019]]}}}}
{{longitem|{{rma|LL06b|tw=4.5em|Jean-Michel Lasry and Pierre-Louis Lions. ''Jeux à champ moyen. II. Horizon fini et contrôle optimal.'' C. R. Math. Acad. Sci. Paris 343 (2006), no. 10, 679–684. [[doi:10.1016/j.crma.2006.09.018]]}}}}
{{longitem|{{rma|LL07|tw=4.5em|Jean-Michel Lasry and Pierre-Louis Lions. ''Mean field games.'' Jpn. J. Math. 2 (2007), no. 1, 229–260. [[doi:10.1007/s11537-007-0657-8]]}}}}
{{longitem|{{rma|GLL11|tw=4.5em|Olivier Guéant, Jean-Michel Lasry, and Pierre-Louis Lions. ''Mean field games and applications.'' Paris-Princeton Lectures on Mathematical Finance 2010, 205–266, Lecture Notes in Math., 2003, Springer, Berlin, 2011. [[doi:10.1007/978-3-642-14660-2_3]]}}}}
{{refend}}
'''Textbooks.'''
{{refbegin}}
{{longitem|{{rma|L82c|tw=4.5em|Pierre-Louis Lions. ''Generalized solutions of Hamilton-Jacobi equations.'' Research Notes in Mathematics, 69. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. iv+317 pp. {{ISBN|0-273-08556-5}}}}}}
{{longitem|{{rma|L96|tw=4.5em|Pierre-Louis Lions. ''Mathematical topics in fluid mechanics. Vol. 1. Incompressible models.'' Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996. xiv+237 pp. {{ISBN|0-19-851487-5}}}}}}
{{longitem|{{rma|L98a|tw=4.5em|Pierre-Louis Lions. ''Mathematical topics in fluid mechanics. Vol. 2. Compressible models.'' Oxford Lecture Series in Mathematics and its Applications, 10. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998. xiv+348 pp. {{ISBN|0-19-851488-3}}}}}}
{{longitem|{{rma|L98b|tw=4.5em|Pierre-Louis Lions. ''On Euler equations and statistical physics.'' Cattedra Galileiana. Scuola Normale Superiore, Classe di Scienze, Pisa, 1998. vi+74 pp.}}}} 
{{longitem|{{rma|CLL98|tw=4.5em|Isabelle Catto, Claude Le Bris, and Pierre-Louis Lions. ''The mathematical theory of thermodynamic limits: Thomas-Fermi type models.'' Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1998. xiv+277 pp. {{ISBN|0-19-850161-7}}}}}}
{{longitem|{{rma|CDLL19|tw=4.5em|Pierre Cardaliaguet, François Delarue, Jean-Michel Lasry, and Pierre-Louis Lions. ''The master equation and the convergence problem in mean field games.'' Annals of Mathematics Studies, 201. Princeton University Press, Princeton, NJ, 2019. x+212 pp. {{ISBN|978-0-691-19071-6}}; 978-0-691-19070-9}}}}
{{refend}}
{{refend}}

==References==
{{reflist}}

==External links==
*[http://www.college-de-france.fr/site/pierre-louis-lions/biographie.htm College de France] his resume at the Collège de France website (in French)
* {{MacTutor Biography|id=Lions}}
* {{MathGenealogy|id=13140}}
* {{IMO results|id=16502}}

{{Commons category|Pierre-Louis Lions}}

{{Fields medalists}}

{{Authority control}}

{{DEFAULTSORT:Lions, Pierre Louis}}
[[Category:1956 births]]
[[Category:Living people]]
[[Category:People from Grasse]]
[[Category:Academic staff of the Collège de France]]
[[Category:20th-century French mathematicians]]
[[Category:21st-century French mathematicians]]
[[Category:Fields Medalists]]
[[Category:French mathematical analysts]]
[[Category:Lycée Louis-le-Grand alumni]]
[[Category:École Normale Supérieure alumni]]
[[Category:Members of the French Academy of Sciences]]
[[Category:Partial differential equation theorists]]
[[Category:International Mathematical Olympiad participants]]
[[Category:Academic staff of Nancy-Université]]
[[Category:Prix Paul Doistau–Émile Blutet laureates]]
[[Category:University of Chicago staff]]