fbpx
Wikipedia

Louis Nirenberg

January 01, 1970

Louis Nirenberg (February 28, 1925 – January 26, 2020) was a Canadian-American mathematician, considered one of the most outstanding mathematicians of the 20th century.[2][3]

Louis Nirenberg
Nirenberg in 1975
Born(1925-02-28)February 28, 1925
Hamilton, Ontario, Canada
DiedJanuary 26, 2020(2020-01-26) (aged 94)
Manhattan, New York, U.S.
CitizenshipCanadian and American
Alma materMcGill University (BS, 1945)
New York University (PhD, 1950)
Known forPartial differential equations
Gagliardo–Nirenberg interpolation inequality
Gagliardo–Nirenberg–Sobolev inequality
Bounded mean oscillation (John–Nirenberg space)
Nirenberg's conjecture[1]
AwardsBôcher Memorial Prize (1959)
Crafoord Prize (1982)
Steele Prize (1994, 2014)
National Medal of Science (1995)
Chern Medal (2010)
Abel Prize in Mathematics (2015)
Scientific career
FieldsMathematics
InstitutionsNew York University
Thesis The determination of a closed convex surface having given line elements  (1949)
Doctoral advisorJames Stoker
Doctoral students
Notes

Nearly all of his work was in the field of partial differential equations. Many of his contributions are now regarded as fundamental to the field, such as his strong maximum principle for second-order parabolic partial differential equations and the Newlander–Nirenberg theorem in complex geometry. He is regarded as a foundational figure in the field of geometric analysis, with many of his works being closely related to the study of complex analysis and differential geometry.[4]

Biography edit

Nirenberg was born in Hamilton, Ontario to Ukrainian Jewish immigrants. He attended Baron Byng High School and McGill University, completing his BS in both mathematics and physics in 1945. Through a summer job at the National Research Council of Canada, he came to know Ernest Courant's wife Sara Paul. She spoke to Courant's father, the eminent mathematician Richard Courant, for advice on where Nirenberg should apply to study theoretical physics. Following their discussion, Nirenberg was invited to enter graduate school at the Courant Institute of Mathematical Sciences at New York University. In 1949, he obtained his doctorate in mathematics, under the direction of James Stoker. In his doctoral work, he solved the "Weyl problem" in differential geometry, which had been a well-known open problem since 1916.

Following his doctorate, he became a professor at the Courant Institute, where he remained for the rest of his career. He was the advisor of 45 PhD students, and published over 150 papers with a number of coauthors, including notable collaborations with Henri Berestycki, Haïm Brezis, Luis Caffarelli, and Yanyan Li, among many others. He continued to carry out mathematical research until the age of 87. On January 26, 2020, Nirenberg died at the age of 94.[5][6][7]

Nirenberg's work was widely recognized, including the following awards and honors:

Mathematical achievements edit

Nirenberg is especially known for his collaboration with Shmuel Agmon and Avron Douglis in which they extended the Schauder theory, as previously understood for second-order elliptic partial differential equations, to the general setting of elliptic systems. With Basilis Gidas and Wei-Ming Ni he made innovative uses of the maximum principle to prove symmetry of many solutions of differential equations. The study of the BMO function space was initiated by Nirenberg and Fritz John in 1961; while it was originally introduced by John in the study of elastic materials, it has also been applied to games of chance known as martingales.[18] His 1982 work with Luis Caffarelli and Robert Kohn made a seminal contribution to the Navier–Stokes existence and smoothness, in the field of mathematical fluid mechanics.

Other achievements include the resolution of the Minkowski problem in two-dimensions, the Gagliardo–Nirenberg interpolation inequality, the Newlander-Nirenberg theorem in complex geometry, and the development of pseudo-differential operators with Joseph Kohn.

Navier-Stokes equations edit

The Navier-Stokes equations were developed in the early 1800s to model the physics of fluid mechanics. Jean Leray, in a seminal achievement in the 1930s, formulated an influential notion of weak solution for the equations and proved their existence.[19] His work was later put into the setting of a boundary value problem by Eberhard Hopf.[20]

A breakthrough came with work of Vladimir Scheffer in the 1970s. He showed that if a smooth solution of the Navier−Stokes equations approaches a singular time, then the solution can be extended continuously to the singular time away from, roughly speaking, a curve in space.[21] Without making such a conditional assumption on smoothness, he established the existence of Leray−Hopf solutions which are smooth away from a two-dimensional surface in spacetime.[22] Such results are referred to as "partial regularity." Soon afterwards, Luis Caffarelli, Robert Kohn, and Nirenberg localized and sharpened Scheffer's analysis.[CKN82] The key tool of Scheffer's analysis was an energy inequality providing localized integral control of solutions. It is not automatically satisfied by Leray−Hopf solutions, but Scheffer and Caffarelli−Kohn−Nirenberg established existence theorems for solutions satisfying such inequalities. With such "a priori" control as a starting point, Caffarelli−Kohn−Nirenberg were able to prove a purely local result on smoothness away from a curve in spacetime, improving Scheffer's partial regularity.

Similar results were later found by Michael Struwe, and a simplified version of Caffarelli−Kohn−Nirenberg's analysis was later found by Fang-Hua Lin.[23][24] In 2014, the American Mathematical Society recognized Caffarelli−Kohn−Nirenberg's paper with the Steele Prize for Seminal Contribution to Research, saying that their work was a "landmark" providing a "source of inspiration for a generation of mathematicians." The further analysis of the regularity theory of the Navier−Stokes equations is, as of 2021, a well-known open problem.

Nonlinear elliptic partial differential equations edit

In the 1930s, Charles Morrey found the basic regularity theory of quasilinear elliptic partial differential equations for functions on two-dimensional domains.[25] Nirenberg, as part of his Ph.D. thesis, extended Morrey's results to the setting of fully nonlinear elliptic equations.[N53a] The works of Morrey and Nirenberg made extensive use of two-dimensionality, and the understanding of elliptic equations with higher-dimensional domains was an outstanding open problem.

The Monge-Ampère equation, in the form of prescribing the determinant of the hessian of a function, is one of the standard examples of a fully nonlinear elliptic equation. In an invited lecture at the 1974 International Congress of Mathematicians, Nirenberg announced results obtained with Eugenio Calabi on the boundary-value problem for the Monge−Ampère equation, based upon boundary regularity estimates and a method of continuity.[26] However, they soon realized their proofs to be incomplete.[26] In 1977, Shiu-Yuen Cheng and Shing-Tung Yau resolved the existence and interior regularity for the Monge-Ampère equation, showing in particular that if the determinant of the hessian of a function is smooth, then the function itself must be smooth as well.[27] Their work was based upon the relation via the Legendre transform to the Minkowski problem, which they had previously resolved by differential-geometric estimates.[28] In particular, their work did not make use of boundary regularity, and their results left such questions unresolved.

In collaboration with Luis Caffarelli and Joel Spruck, Nirenberg resolved such questions, directly establishing boundary regularity and using it to build a direct approach to the Monge−Ampère equation based upon the method of continuity.[CNS84] Calabi and Nirenberg had successfully demonstrated uniform control of the first two derivatives; the key for the method of continuity is the more powerful uniform Hölder continuity of the second derivatives. Caffarelli, Nirenberg, and Spruck established a delicate version of this along the boundary,[29] which they were able to establish as sufficient by using Calabi's third-derivative estimates in the interior.[30] With Joseph Kohn, they found analogous results in the setting of the complex Monge−Ampère equation.[C+85] In such general situations, the Evans−Krylov theory[29] is a more flexible tool than the computation-based calculations of Calabi.

Caffarelli, Nirenberg, and Spruck were able to extend their methods to more general classes of fully nonlinear elliptic partial differential equations, in which one studies functions for which certain relations between the hessian's eigenvalues are prescribed.[CNS85] As a particular case of their new class of equations, they were able to partially resolve the boundary-value problem for special Lagrangians.

Linear elliptic systems edit

Nirenberg's most renowned work from the 1950s deals with "elliptic regularity." With Avron Douglis, Nirenberg extended the Schauder estimates, as discovered in the 1930s in the context of second-order elliptic equations, to general elliptic systems of arbitrary order.[DN55] In collaboration with Shmuel Agmon and Douglis, Nirenberg proved boundary regularity for elliptic equations of arbitrary order.[ADN59] They later extended their results to elliptic systems of arbitrary order.[ADN64] With Morrey, Nirenberg proved that solutions of elliptic systems with analytic coefficients are themselves analytic, extending to the boundary earlier known work.[MN57] These contributions to elliptic regularity are now considered as part of a "standard package" of information, and are covered in many textbooks. The Douglis−Nirenberg and Agmon−Douglis−Nirenberg estimates, in particular, are among the most widely-used tools in elliptic partial differential equations.[31]

With Yanyan Li, and motivated by composite materials in elasticity theory, Nirenberg studied linear elliptic systems in which the coefficients are Hölder continuous in the interior but possibly discontinuous on the boundary. Their result is that the gradient of the solution is Hölder continuous, with a L∞ estimate for the gradient which is independent of the distance from the boundary.[LN03]

Maximum principle and its applications edit

In the case of harmonic functions, the maximum principle was known in the 1800s, and was used by Carl Friedrich Gauss.[32][33] In the early 1900s, complicated extensions to general second-order elliptic partial differential equations were found by Sergei Bernstein, Leon Lichtenstein, and Émile Picard; it was not until the 1920s that the simple modern proof was found by Eberhard Hopf.[34] In one of his earliest works, Nirenberg adapted Hopf's proof to second-order parabolic partial differential equations, thereby establishing the strong maximum principle in that context.[N53b] As in the earlier work, such a result had various uniqueness and comparison theorems as corollaries. Nirenberg's work is now regarded as one of the foundations of the field of parabolic partial differential equations, and is ubiquitous across the standard textbooks.[35][36][37][38][39][40]

In the 1950s, A.D. Alexandrov introduced an elegant "moving plane" reflection method, which he used as the context for applying the maximum principle to characterize the standard sphere as the only closed hypersurface of Euclidean space with constant mean curvature. In 1971, James Serrin utilized Alexandrov's technique to prove that highly symmetric solutions of certain second-order elliptic partial differential equations must be supported on symmetric domains. Nirenberg realized that Serrin's work could be reformulated so as to prove that solutions of second-order elliptic partial differential equations inherit symmetries of their domain and of the equation itself. Such results do not hold automatically, and it is nontrivial to identify which special features of a given problem are relevant. For example, there are many harmonic functions on Euclidean space which fail to be rotationally symmetric, despite the rotational symmetry of the Laplacian and of Euclidean space.

Nirenberg's first results on this problem were obtained in collaboration with Basilis Gidas and Wei-Ming Ni. They developed a precise form of Alexandrov and Serrin's technique, applicable even to fully nonlinear elliptic and parabolic equations.[GNN79] In a later work, they developed a version of the Hopf lemma applicable on unbounded domains, thereby improving their work in the case of equations on such domains.[GNN81] Their main applications deal with rotational symmetry. Due to such results, in many cases of geometric or physical interest, it is sufficient to study ordinary differential equations rather than partial differential equations.

Later, with Henri Berestycki, Nirenberg used the Alexandrov−Bakelman−Pucci estimate[29] to improve and modify the methods of Gidas−Ni−Nirenberg, significantly reducing the need to assume regularity of the domain.[BN91a] In an important result with Srinivasa Varadhan, Berestycki and Nirenberg continued the study of domains with no assumed regularity. For linear operators, they related the validity of the maximum principle to positivity of a first eigenvalue and existence of a first eigenfunction.[BNV94] With Luis Caffarelli, Berestycki and Nirenberg applied their results to symmetry of functions on cylindrical domains.[BCN96] They obtained in particular a partial resolution of a well-known conjecture of Ennio De Giorgi on translational symmetry, which was later fully resolved in Ovidiu Savin's Ph.D. thesis.[BCN97b][41][42] They further applied their method to obtain qualitative phenomena on general unbounded domains, extending earlier works of Maria Esteban and Pierre-Louis Lions.[BCN97a]

Functional inequalities edit

Nirenberg and Emilio Gagliardo independently proved fundamental inequalities for Sobolev spaces, now known as the Gagliardo–Nirenberg–Sobolev inequality and the Gagliardo–Nirenberg interpolation inequalities.[N59] They are used ubiquitously throughout the literature on partial differential equations; as such, it has been of great interest to extend and adapt them to various situations. Nirenberg himself would later clarify the possible exponents which can appear in the interpolation inequality.[N66] With Luis Caffarelli and Robert Kohn, Nirenberg would establish corresponding inequalities for certain weighted norms.[CKN84] Caffarelli, Kohn, and Nirenberg's norms were later investigated more fully in notable work by Florin Catrina and Zhi-Qiang Wang.[43]

Immediately following Fritz John's introduction of the bounded mean oscillation (BMO) function space in the theory of elasticity, he and Nirenberg gave a further study of the space, proving in particular the "John−Nirenberg inequality," which constrains the size of the set on which a BMO function is far from its average value.[JN61] Their work, which is an application of the Calderon−Zygmund decomposition, has become a part of the standard mathematical literature. Expositions are contained in standard textbooks on probability,[44] complex analysis,[45] harmonic analysis,[46] Fourier analysis,[47] and partial differential equations.[29] Among other applications, it is particularly fundamental to Jürgen Moser's Harnack inequality and subsequent work.[48][49][29]

The John−Nirenberg inequality and the more general foundations of the BMO theory were worked out by Nirenberg and Haïm Brézis in the context of maps between Riemannian manifolds.[BN95] Among other results, they were able to establish that smooth maps which are close in BMO norm have the same topological degree, and hence that degree can be meaningfully defined for mappings of vanishing mean oscillation (VMO).

Calculus of variations edit

In the setting of topological vector spaces, Ky Fan developed a minimax theorem with applications in game theory.[50][51] With Haïm Brezis and Guido Stampacchia, Nirenberg derived results extending both Fan's theory and Stampacchia's generalization of the Lax-Milgram theorem.[BNS72][52] Their work has applications to the subject of variational inequalities.[53]

By adapting the Dirichlet energy, it is standard to recognize solutions of certain wave equations as critical points of functionals. With Brezis and Jean-Michel Coron, Nirenberg found a novel functional whose critical points can be directly used to construct solutions of wave equations.[BCN80] They were able to apply the mountain pass theorem to their new functional, thereby establishing the existence of periodic solutions of certain wave equations, extending a result of Paul Rabinowitz.[54] Part of their work involved small extensions of the standard mountain pass theorem and Palais-Smale condition, which have become standard in textbooks.[55][56][57] In 1991, Brezis and Nirenberg showed how Ekeland's variational principle could be applied to extend the mountain pass theorem, with the effect that almost-critical points can be found without requiring the Palais−Smale condition.[BN91b][57]

A fundamental contribution of Brezis and Nirenberg to critical point theory dealt with local minimizers.[BN93] In principle, the choice of function space is highly relevant, and a function could minimize among smooth functions without minimizing among the broader class of Sobolev functions. Making use of an earlier regularity result of Brezis and Tosio Kato, Brezis and Nirenberg ruled out such phenomena for a certain class of Dirichlet-type functionals.[58] Their work was later extended by Jesús García Azorero, Juan Manfredi, and Ireneo Peral.[59]

In one of Nirenberg's most widely cited papers, he and Brézis studied the Dirichlet problem for Yamabe-type equations on Euclidean spaces, following part of Thierry Aubin's work on the Yamabe problem.[BN83] With Berestycki and Italo Capuzzo-Dolcetta, Nirenberg studied superlinear equations of Yamabe type, giving various existence and non-existence results.[BCN94]

Nonlinear functional analysis edit

Agmon and Nirenberg made an extensive study of ordinary differential equations in Banach spaces, relating asymptotic representations and the behavior at infinity of solutions to

 

to the spectral properties of the operator A. Applications include the study of rather general parabolic and elliptic-parabolic problems.[AN63]

Brezis and Nirenberg gave a study of the perturbation theory of nonlinear perturbations of noninvertible transformations between Hilbert spaces; applications include existence results for periodic solutions of some semilinear wave equations.[BN78a][BN78b]

In John Nash's work on the isometric embedding problem, the key step is a small perturbation result, highly reminiscent of an implicit function theorem; his proof used a novel combination of Newton's method (in an infinitesimal form) with smoothing operators.[60] Nirenberg was one of many mathematicians to put Nash's ideas into systematic and abstract frameworks, referred to as Nash-Moser theorems. Nirenberg's formulation is particularly simple, isolating the basic analytic ideas underlying the analysis of most Nash-Moser iteration schemes.[N72] Within a similar framework, he proved an abstract form of the Cauchy–Kowalevski theorem, as a particular case of a theorem on solvability of ordinary differential equations in families of Banach spaces.[N72] His work was later simplified by Takaaki Nishida and used in an analysis of the Boltzmann equation.[61][62]

Geometric problems edit

Making use of his work on fully nonlinear elliptic equations[N53a], Nirenberg's Ph.D. thesis provided a resolution of the Weyl problem and Minkowski problem in the field of differential geometry.[N53c] The former asks for the existence of isometric embeddings of positively curved Riemannian metrics on the two-dimensional sphere into three-dimensional Euclidean space, while the latter asks for closed surfaces in three-dimensional Euclidean space for which the Gauss map prescribes the Gaussian curvature. The key is that the "Darboux equation" from surface theory is of Monge−Ampère type, so that Nirenberg's regularity theory becomes useful in the method of continuity. John Nash's well-known isometric embedding theorems, established soon afterwards, have no apparent relation to the Weyl problem, which deals simultaneously with high-regularity embeddings and low codimension.[63][60] Nirenberg's work on the Minkowski problem was extended to Riemannian settings by Aleksei Pogorelov. In higher dimensions, the Minkowski problem was resolved by Shiu-Yuen Cheng and Shing-Tung Yau.[28] Other approaches to the Minkowski problem have developed from Caffarelli, Nirenberg, and Spruck's fundamental contributions to the theory of nonlinear elliptic equations.[CNS85]

In one of his very few articles not centered on analysis, Nirenberg and Philip Hartman characterized the cylinders in Euclidean space as the only complete hypersurfaces which are intrinsically flat.[HN59] This can also be viewed as resolving a question on the isometric embedding of flat manifolds as hypersurfaces. Such questions and natural generalizations were later taken up by Cheng, Yau, and Harold Rosenberg, among others.[64][65]

Answering a question posed to Nirenberg by Shiing-Shen Chern and André Weil, Nirenberg and his doctoral student August Newlander proved what is now known as the Newlander-Nirenberg theorem, which provides the precise algebraic condition under which an almost complex structure arises from a holomorphic coordinate atlas.[NN57] The Newlander-Nirenberg theorem is now considered as a foundational result in complex geometry, although the result itself is far better known than the proof, which is not usually covered in introductory texts, as it relies on advanced methods in partial differential equations. Nirenberg and Joseph Kohn, following earlier work by Kohn, studied the ∂-Neumann problem on pseudoconvex domains, and demonstrated the relation of the regularity theory to the existence of subelliptic estimates for the ∂ operator.[KN65b]

The classical Poincaré disk model assigns the metric of hyperbolic space to the unit ball. Nirenberg and Charles Loewner studied the more general means of naturally assigning a complete Riemannian metric to bounded open subsets of Euclidean space.[LN74] Geometric calculations show that solutions of certain semilinear Yamabe-type equations can be used to define metrics of constant scalar curvature, and that the metric is complete if the solution diverges to infinity near the boundary. Loewner and Nirenberg established existence of such solutions on certain domains. Similarly, they studied a certain Monge−Ampère equation with the property that, for any negative solution extending continuously to zero at the boundary, one can define a complete Riemannian metric via the hessian. These metrics have the special property of projective invariance, so that projective transformation from one given domain to another becomes an isometry of the corresponding metrics.

Pseudo-differential operators edit

Joseph Kohn and Nirenberg introduced the notion of pseudo-differential operators.[KN65a] Nirenberg and François Trèves investigated the famous Lewy's example for a non-solvable linear PDE of second order, and discovered the conditions under which it is solvable, in the context of both partial differential operators and pseudo-differential operators.[NT63][NT70] Their introduction of local solvability conditions with analytic coefficients has become a focus for researchers such as R. Beals, C. Fefferman, R.D. Moyer, Lars Hörmander, and Nils Dencker who solved the pseudo-differential condition for Lewy's equation. This opened up further doors into the local solvability of linear partial differential equations.

Major publications edit

Books and surveys.

N73.
Nirenberg, Louis (1973). Lectures on linear partial differential equations. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics. Vol. 17. Providence, RI: American Mathematical Society. doi:10.1090/cbms/017. ISBN 978-0-8218-1667-7. MR 0450755. Zbl 0267.35001.
N81.
Nirenberg, L. (1981). "Variational and topological methods in nonlinear problems". Bulletin of the American Mathematical Society. New Series. 4 (3): 267–302. doi:10.1090/S0273-0979-1981-14888-6. MR 0609039. Zbl 0468.47040.
N94.
Nirenberg, Louis (1994). "Partial differential equations in the first half of the century". In Pier, Jean-Paul (ed.). Development of Mathematics 1900–1950. Symposium held in Luxembourg, June 1992. Basel: Birkhäuser. pp. 479–515. ISBN 3-7643-2821-5. MR 1298641. Zbl 0807.01017.
N01.
Nirenberg, Louis (2001). Topics in nonlinear functional analysis. Courant Lecture Notes in Mathematics. Vol. 6. Chapter 6 by E. Zehnder. Notes by R. A. Artino (Revised reprint of the 1974 original ed.). New York: New York University, Courant Institute of Mathematical Sciences. doi:10.1090/cln/006. ISBN 0-8218-2819-3. MR 0488102. Zbl 0286.47037.
N18.
Nirenberg, Louis (2018). Lectures on differential equations and differential geometry. Classical Topics in Mathematics. Vol. 7. With a preface by Shiu-Yuen Cheng and Lizhen Ji. Beijing: Higher Education Press. ISBN 978-7-04-050302-9. MR 3889014. Zbl 1465.35001.

Articles.

N53a.
Nirenberg, Louis (1953). "On nonlinear elliptic partial differential equations and Hölder continuity". Communications on Pure and Applied Mathematics. 6 (1): 103–156. doi:10.1002/cpa.3160060105. MR 0064986. Zbl 0050.09801. (Erratum: doi:10.1002/cpa.3160060304)
N53b.
Nirenberg, Louis (1953). "A strong maximum principle for parabolic equations". Communications on Pure and Applied Mathematics. 6 (2): 167–177. doi:10.1002/cpa.3160060202. MR 0055544. Zbl 0050.09601.
N53c.
Nirenberg, Louis (1953). "The Weyl and Minkowski problems in differential geometry in the large". Communications on Pure and Applied Mathematics. 6 (3): 337–394. doi:10.1002/cpa.3160060303. MR 0058265. Zbl 0051.12402.
DN55.
Douglis, Avron; Nirenberg, Louis (1955). "Interior estimates for elliptic systems of partial differential equations". Communications on Pure and Applied Mathematics. 8 (4): 503–538. doi:10.1002/cpa.3160080406. MR 0075417. Zbl 0066.08002.
N55.
Nirenberg, Louis (1955). "Remarks on strongly elliptic partial differential equations". Communications on Pure and Applied Mathematics. 8 (4): 649–675. doi:10.1002/cpa.3160080414. MR 0075415. Zbl 0067.07602.
MN57.
Morrey, C. B. Jr.; Nirenberg, L. (1957). "On the analyticity of the solutions of linear elliptic systems of partial differential equations". Communications on Pure and Applied Mathematics. 10 (2): 271–290. doi:10.1002/cpa.3160100204. MR 0089334. Zbl 0082.09402.
NN57.
Newlander, A.; Nirenberg, L. (1957). "Complex analytic coordinates in almost complex manifolds". Annals of Mathematics. Second Series. 65 (3): 391–404. doi:10.2307/1970051. JSTOR 1970051. MR 0088770. Zbl 0079.16102.
ADN59.
Agmon, S.; Douglis, A.; Nirenberg, L. (1959). "Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I". Communications on Pure and Applied Mathematics. 12 (4): 623–727. doi:10.1002/cpa.3160120405. MR 0125307. Zbl 0093.10401.
HN59.
Hartman, Philip; Nirenberg, Louis (1959). "On spherical image maps whose Jacobians do not change sign". American Journal of Mathematics. 81 (4): 901–920. doi:10.2307/2372995. JSTOR 2372995. MR 0126812. Zbl 0094.16303.
N59.
Nirenberg, L. (1959). "On elliptic partial differential equations". Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie III. 13 (2): 115–162. MR 0109940. Zbl 0088.07601.
JN61.
John, F.; Nirenberg, L. (1961). "On functions of bounded mean oscillation". Communications on Pure and Applied Mathematics. 14 (3): 415–426. doi:10.1002/cpa.3160140317. MR 0131498. Zbl 0102.04302.
AN63.
Agmon, S.; Nirenberg, L. (1963). "Properties of solutions of ordinary differential equations in Banach space". Communications on Pure and Applied Mathematics. 16 (2): 121–239. doi:10.1002/cpa.3160160204. MR 0155203. Zbl 0117.10001.
NT63.
Nirenberg, L.; Treves, F. (1963). "Solvability of a first order linear partial differential equation". Communications on Pure and Applied Mathematics. 16 (3): 331–351. doi:10.1002/cpa.3160160308. MR 0163045. Zbl 0117.06104.
ADN64.
Agmon, S.; Douglis, A.; Nirenberg, L. (1964). "Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II". Communications on Pure and Applied Mathematics. 17 (1): 35–92. doi:10.1002/cpa.3160170104. MR 0162050. Zbl 0123.28706.
KN65a.
Kohn, J. J.; Nirenberg, L. (1965). "An algebra of pseudo-differential operators". Communications on Pure and Applied Mathematics. 18 (1–2): 269–305. doi:10.1002/cpa.3160180121. MR 0176362. Zbl 0171.35101.
KN65b.
Kohn, J. J.; Nirenberg, L. (1965). "Non-coercive boundary value problems". Communications on Pure and Applied Mathematics. 18 (3): 443–492. doi:10.1002/cpa.3160180305. MR 0181815. Zbl 0125.33302.
N66.
Nirenberg, L. (1966). "An extended interpolation inequality". Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie III. 20 (4): 733–737. MR 0208360. Zbl 0163.29905.
KN67.
Kohn, J. J.; Nirenberg, L. (1967). "Degenerate elliptic-parabolic equations of second order". Communications on Pure and Applied Mathematics. 20 (4): 797–872. doi:10.1002/cpa.3160200410. MR 0234118. Zbl 0153.14503.
NT70.
Nirenberg, Louis; Trèves, François (1970). "On local solvability of linear partial differential equations. I. Necessary conditions". Communications on Pure and Applied Mathematics. 23 (1): 1–38. doi:10.1002/cpa.3160230102. MR 0264470. Zbl 0191.39103.
BNS72.
Brézis, H.; Nirenberg, L.; Stampacchia, G. (1972). "A remark on Ky Fan's minimax principle". Bollettino della Unione Matematica Italiana. Series IV. 6: 293–300. MR 0324498. Zbl 0264.49013.
N72.
Nirenberg, L. (1972). "An abstract form of the nonlinear Cauchy–Kowalewski theorem". Journal of Differential Geometry. 6 (4): 561–576. doi:10.4310/jdg/1214430643. MR 0322321. Zbl 0257.35001.
NW73.
Nirenberg, Louis; Walker, Homer F. (1973). "The null spaces of elliptic partial differential operators in Rn". Journal of Mathematical Analysis and Applications. 42 (2): 271–301. doi:10.1016/0022-247X(73)90138-8. MR 0320821. Zbl 0272.35029.
LN74.
Loewner, Charles; Nirenberg, Louis (1974). "Partial differential equations invariant under conformal or projective transformations". In Ahlfors, Lars V.; Kra, Irwin; Maskit, Bernard; Nirenberg, Louis (eds.). Contributions to Analysis: A Collection of Papers Dedicated to Lipman Bers. New York: Academic Press. pp. 245–272. doi:10.1016/B978-0-12-044850-0.50027-7. MR 0358078. Zbl 0298.35018.
KN77.
Kinderlehrer, D.; Nirenberg, L. (1977). "Regularity in free boundary problems". Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV. 4 (2): 373–391. MR 0440187. Zbl 0352.35023.
BN78a.
Brézis, H.; Nirenberg, L. (1978). "Characterizations of the ranges of some nonlinear operators and applications to boundary value problems". Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV. 5 (2): 225–326. MR 0513090. Zbl 0386.47035.
BN78b.
Brézis, H.; Nirenberg, L. (1978). "Forced vibrations for a nonlinear wave equation". Communications on Pure and Applied Mathematics. 31 (1): 1–30. doi:10.1002/cpa.3160310102. MR 0470377. Zbl 0378.35040.
GNN79.
Gidas, B.; Ni, Wei Ming; Nirenberg, L. (1979). "Symmetry and related properties via the maximum principle". Communications in Mathematical Physics. 68 (3): 209–243. Bibcode:1979CMaPh..68..209G. doi:10.1007/BF01221125. MR 0544879. S2CID 56251822. Zbl 0425.35020.
BCN80.
Brézis, Haïm; Coron, Jean-Michel; Nirenberg, Louis (1980). "Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz". Communications on Pure and Applied Mathematics. 33 (5): 667–684. doi:10.1002/cpa.3160330507. MR 0586417. Zbl 0484.35057.
GNN81.
Gidas, B.; Ni, Wei Ming; Nirenberg, L. (1981). "Symmetry of positive solutions of nonlinear elliptic equations in Rn". In Nachbin, Leopoldo (ed.). Mathematical Analysis and Applications. Part A. Essays dedicated to Laurent Schwartz on the occasion of his 65th birthday. Advances in Mathematics, Supplementary Studies. Vol. 7a. New York–London: Academic Press. pp. 369–402. ISBN 0-12-512801-0. MR 0634248. Zbl 0469.35052.
CKN82.
Caffarelli, L.; Kohn, R.; Nirenberg, L. (1982). "Partial regularity of suitable weak solutions of the Navier–Stokes equations". Communications on Pure and Applied Mathematics. 35 (6): 771–831. Bibcode:1982CPAM...35..771C. doi:10.1002/cpa.3160350604. MR 0673830. Zbl 0509.35067.
BN83.
Brézis, Haïm; Nirenberg, Louis (1983). "Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents". Communications on Pure and Applied Mathematics. 36 (4): 437–477. doi:10.1002/cpa.3160360405. MR 0709644. Zbl 0541.35029.
CKN84.
Caffarelli, L.; Kohn, R.; Nirenberg, L. (1984). "First order interpolation inequalities with weights". Compositio Mathematica. 53 (3): 259–275. MR 0768824. Zbl 0563.46024.
CNS84.
Caffarelli, L.; Nirenberg, L.; Spruck, J. (1984). "The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge–Ampère equation". Communications on Pure and Applied Mathematics. 37 (3): 369–402. doi:10.1002/cpa.3160370306. MR 0739925. Zbl 0598.35047. (Erratum: doi:10.1002/cpa.3160400508)
C+85.
Caffarelli, L.; Kohn, J. J.; Nirenberg, L.; Spruck, J. (1985). "The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge–Ampère, and uniformly elliptic, equations". Communications on Pure and Applied Mathematics. 38 (2): 209–252. doi:10.1002/cpa.3160380206. MR 0780073. Zbl 0598.35048.
CNS85.
Caffarelli, L.; Nirenberg, L.; Spruck, J. (1985). "The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian". Acta Mathematica. 155 (3–4): 261–301. doi:10.1007/BF02392544. MR 0806416. Zbl 0654.35031.
BN91a.
Berestycki, H.; Nirenberg, L. (1991). "On the method of moving planes and the sliding method". Boletim da Sociedade Brasileira de Matemática. New Series. 22 (1): 1–37. doi:10.1007/BF01244896. MR 1159383. S2CID 119878360. Zbl 0784.35025.
BN91b.
Brezis, Haïm; Nirenberg, Louis (1991). "Remarks on finding critical points". Communications on Pure and Applied Mathematics. 44 (8–9): 939–963. doi:10.1002/cpa.3160440808. MR 1127041. Zbl 0751.58006.
BN92.
Berestycki, Henri; Nirenberg, Louis (1992). "Travelling fronts in cylinders". Annales de l'Institut Henri Poincaré C. 9 (5): 497–572. Bibcode:1992AIHPC...9..497B. doi:10.1016/S0294-1449(16)30229-3. MR 1191008. Zbl 0799.35073.
BN93.
Brezis, Haïm; Nirenberg, Louis (1993). "H1 versus C1 local minimizers". Comptes Rendus de l'Académie des Sciences, Série I. 317 (5): 465–472. MR 1239032. Zbl 0803.35029.
BCN94.
Berestycki, H.; Capuzzo-Dolcetta, I.; Nirenberg, L. (1994). "Superlinear indefinite elliptic problems and nonlinear Liouville theorems". Topological Methods in Nonlinear Analysis. 4 (1): 59–78. doi:10.12775/TMNA.1994.023. MR 1321809. Zbl 0816.35030.
BNV94.
Berestycki, H.; Nirenberg, L.; Varadhan, S. R. S. (1994). "The principal eigenvalue and maximum principle for second-order elliptic operators in general domains". Communications on Pure and Applied Mathematics. 47 (1): 47–92. doi:10.1002/cpa.3160470105. MR 1258192. Zbl 0806.35129.
BCN95.
Berestycki, Henri; Capuzzo-Dolcetta, Italo; Nirenberg, Louis (1995). "Variational methods for indefinite superlinear homogeneous elliptic problems". Nonlinear Differential Equations and Applications. 2 (4): 553–572. doi:10.1007/BF01210623. MR 1356874. S2CID 119668931. Zbl 0840.35035.
BN95.
Brezis, H.; Nirenberg, L. (1995). "Degree theory and BMO. I. Compact manifolds without boundaries". Selecta Mathematica. New Series. 1 (2): 197–263. doi:10.1007/BF01671566. MR 1354598. S2CID 195270732. Zbl 0852.58010.
BCN96.
Berestycki, H.; Caffarelli, L. A.; Nirenberg, L. (1996). "Inequalities for second-order elliptic equations with applications to unbounded domains. I". Duke Mathematical Journal. 81 (2): 467–494. doi:10.1215/S0012-7094-96-08117-X. MR 1395408. Zbl 0860.35004.
BCN97a.
Berestycki, H.; Caffarelli, L. A.; Nirenberg, L. (1997). "Monotonicity for elliptic equations in unbounded Lipschitz domains". Communications on Pure and Applied Mathematics. 50 (11): 1089–1111. doi:10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6. MR 1470317. Zbl 0906.35035.
BCN97b.
Berestycki, Henri; Caffarelli, Luis; Nirenberg, Louis (1997). "Further qualitative properties for elliptic equations in unbounded domains". Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV. 25 (1–2): 69–94. MR 1655510. Zbl 1079.35513.
LN03.
Li, Yanyan; Nirenberg, Louis (2003). "Estimates for elliptic systems from composite material". Communications on Pure and Applied Mathematics. 56 (7): 892–925. doi:10.1002/cpa.10079. MR 1990481. S2CID 122040650. Zbl 1125.35339.

References edit

  1. ^ Lawson, H. Blaine Jr. (April 21, 2012). "Reflections on the Early Mathematical Life of Bob Osserman" (PDF).
  2. ^ Allyn Jackson (March 2002). (PDF). Notices of the AMS. 49 (4): 441–449. Archived from the original (PDF) on March 3, 2016. Retrieved March 26, 2015.
  3. ^ Caffarelli, Luis A.; Li, YanYan. Preface [Dedicated to Louis Nirenberg on the occasion of his 85th birthday. Part I]. Discrete Contin. Dyn. Syst. 28 (2010), no. 2, i–ii. doi:10.3934/dcds.2010.28.2i
  4. ^ Yau, Shing-Tung. Perspectives on geometric analysis. Surveys in differential geometry. Vol. X, 275–379, Surv. Differ. Geom., 10, Int. Press, Somerville, MA, 2006.
  5. ^ Morto il grande matematico Louis Nirenberg (in Italian)
  6. ^ Chang, Kenneth (January 31, 2020). "Louis Nirenberg, 'One of the Great Mathematicians,' Dies at 94". New York Times. Retrieved February 19, 2020.
  7. ^ Shields, Brit; Barany, Michael J. (February 17, 2020). "Louis Nirenberg (1925–2020)". Nature. Retrieved February 19, 2020.
  8. ^ "Browse Prizes and Awards". American Mathematical Society. Retrieved August 12, 2022.
  9. ^ "Louis Nirenberg". American Academy of Arts & Sciences. Retrieved May 5, 2022.
  10. ^ "Louis Nirenberg". www.nasonline.org. Retrieved May 5, 2022.
  11. ^ "The Crafoord Prize 1982". The Crafoord Prize. May 25, 1982.
  12. ^ "Jeffery-Williams Prize". CMS-SMC. Retrieved August 12, 2022.
  13. ^ "APS Member History". search.amphilsoc.org. Retrieved May 5, 2022.
  14. ^ 1994 Steele Prizes. Notices Amer. Math. Soc. 41 (1994), no. 8, 905–912.
  15. ^ Louis Nirenberg receives National Medal of Science. With contributions by Luis Caffarelli and Joseph J. Kohn. Notices Amer. Math. Soc. 43 (1996), no. 10, 1111–1116.
  16. ^ 2010 Chern Medal awarded. Notices Amer. Math. Soc. 57 (2010), no. 11, 1472–1474.
  17. ^ "2015: John F. Nash and Louis Nirenberg". The Norwegian Academy of Science and Letters. Retrieved August 12, 2022.
  18. ^ "John F. Nash Jr. and Louis Nirenberg share the Abel Prize". The Abel Prize. March 25, 2015. Retrieved March 26, 2015.
  19. ^ Leray, Jean. Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math. 63 (1934), no. 1, 193–248.
  20. ^ Hopf, Eberhard. Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4 (1951), 213–231.
  21. ^ Scheffer, Vladimir. Partial regularity of solutions to the Navier-Stokes equations. Pacific J. Math. 66 (1976), no. 2, 535–552.
  22. ^ Scheffer, Vladimir. Hausdorff measure and the Navier-Stokes equations. Comm. Math. Phys. 55 (1977), no. 2, 97–112.
  23. ^ Struwe, Michael. On partial regularity results for the Navier-Stokes equations. Comm. Pure Appl. Math. 41 (1988), no. 4, 437–458.
  24. ^ Lin, Fanghua. A new proof of the Caffarelli-Kohn-Nirenberg theorem. Comm. Pure Appl. Math. 51 (1998), no. 3, 241–257.
  25. ^ Morrey, Charles B., Jr. On the solutions of quasi-linear elliptic partial differential equations. Trans. Amer. Math. Soc. 43 (1938), no. 1, 126–166.
  26. ^ a b See the second page of [CNS84].
  27. ^ Cheng, Shiu Yuen; Yau, Shing Tung. On the regularity of the Monge-Ampère equation det(∂2u/∂xi∂xj) = F(x,u). Comm. Pure Appl. Math. 30 (1977), no. 1, 41–68.
  28. ^ a b Cheng, Shiu Yuen; Yau, Shing Tung. On the regularity of the solution of the n-dimensional Minkowski problem. Comm. Pure Appl. Math. 29 (1976), no. 5, 495–516.
  29. ^ a b c d e Gilbarg, David; Trudinger, Neil S. Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. xiv+517 pp. ISBN 3-540-41160-7
  30. ^ Calabi, Eugenio. Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens. Michigan Math. J. 5 (1958), 105–126.
  31. ^ Morrey, Charles B., Jr. Multiple integrals in the calculus of variations. Die Grundlehren der mathematischen Wissenschaften, Band 130 Springer-Verlag New York, Inc., New York 1966 ix+506 pp.
  32. ^ The historical comments and references are taken from James Serrin's commentary on page 9 of Selected Works of Eberhard Hopf with Commentaries. American Mathematical Society, Providence, RI, 2002. xxiv+386 pp.
  33. ^ Gauss, C.F. Allgemeine Theorie des Erdmagnetismus. Resultate aus den Beobachtungen des Magnetischen Vereins im Jahre 1838.
  34. ^ Hopf, Eberhard. Elementare Bemerkungen über die Lösungen partieller Differentialgleichngen zweiter Ordnung vom elliptischen Typus (1927)
  35. ^ Evans, Lawrence C. Partial differential equations. Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. xxii+749 pp.
  36. ^ Friedman, Avner. Partial differential equations of parabolic type. Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964 xiv+347 pp.
  37. ^ Ladyženskaja, O.A.; Solonnikov, V.A.; Uralʹceva, N.N. Linear and quasilinear equations of parabolic type. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1968 xi+648 pp.
  38. ^ Lieberman, Gary M. Second order parabolic differential equations. World Scientific Publishing Co., Inc., River Edge, NJ, 1996. xii+439 pp.
  39. ^ Protter, Murray H.; Weinberger, Hans F. Maximum principles in differential equations. Corrected reprint of the 1967 original. Springer-Verlag, New York, 1984. x+261 pp.
  40. ^ Smoller, Joel. Shock waves and reaction-diffusion equations. Second edition. Grundlehren der mathematischen Wissenschaften, 258. Springer-Verlag, New York, 1994. xxiv+632 pp.
  41. ^ Savin, Ovidiu. Regularity of flat level sets in phase transitions. Ann. of Math. (2) 169 (2009), no. 1, 41–78.
  42. ^ del Pino, Manuel; Kowalczyk, Michał; Wei, Juncheng. On De Giorgi's conjecture in dimension N≥9. Ann. of Math. (2) 174 (2011), no. 3, 1485–1569.
  43. ^ Catrina, Florin; Wang, Zhi-Qiang. On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions. Comm. Pure Appl. Math. 54 (2001), no. 2, 229–258.
  44. ^ Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der mathematischen Wissenschaften, 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN 3-540-64325-7
  45. ^ Garnett, John B. Bounded analytic functions. Revised first edition. Graduate Texts in Mathematics, 236. Springer, New York, 2007. xiv+459 pp. ISBN 978-0-387-33621-3, 0-387-33621-4
  46. ^ García-Cuerva, José; Rubio de Francia, José L. Weighted norm inequalities and related topics. North-Holland Mathematics Studies, 116. Notas de Matemática, 104. North-Holland Publishing Co., Amsterdam, 1985. x+604 pp. ISBN 0-444-87804-1
  47. ^ Grafakos, Loukas. Modern Fourier analysis. Third edition. Graduate Texts in Mathematics, 250. Springer, New York, 2014. xvi+624 pp. ISBN 978-1-4939-1229-2, 978-1-4939-1230-8
  48. ^ Moser, Jürgen On Harnack's theorem for elliptic differential equations. Comm. Pure Appl. Math. 14 (1961), 577–591.
  49. ^ Moser, Jürgen. A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 17 (1964), 101–134.
  50. ^ Fan, Ky. A generalization of Tychonoff's fixed point theorem. Math. Ann. 142 (1960), 305–310.
  51. ^ Fan, Ky. A minimax inequality and applications. Inequalities, III (Proc. Third Sympos., Univ. California, Los Angeles, Calif., 1969; dedicated to the memory of Theodore S. Motzkin), pp. 103–113. Academic Press, New York, 1972.
  52. ^ Stampacchia, Guido. Formes bilinéaires coercitives sur les ensembles convexes. C. R. Acad. Sci. Paris 258 (1964), 4413–4416.
  53. ^ Aubin, Jean-Pierre; Ekeland, Ivar. Applied nonlinear analysis. Reprint of the 1984 original. Dover Publications, Inc., Mineola, NY, 2006. x+518 pp.
  54. ^ Rabinowitz, Paul H. Free vibrations for a semilinear wave equation. Comm. Pure Appl. Math. 31 (1978), no. 1, 31–68.
  55. ^ Mawhin, Jean; Willem, Michel. Critical point theory and Hamiltonian systems. Applied Mathematical Sciences, 74. Springer-Verlag, New York, 1989. xiv+277 pp.
  56. ^ Struwe, Michael. Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems. Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 34. Springer-Verlag, Berlin, 2008. xx+302 pp.
  57. ^ a b Willem, Michel. Minimax theorems. Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. x+162 pp.
  58. ^ Brézis, Haïm; Kato, Tosio. Remarks on the Schrödinger operator with singular complex potentials. J. Math. Pures Appl. (9) 58 (1979), no. 2, 137–151.
  59. ^ García Azorero, J. P.; Peral Alonso, I.; Manfredi, Juan J. Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations. Commun. Contemp. Math. 2 (2000), no. 3, 385–404.
  60. ^ a b Nash, John. The imbedding problem for Riemannian manifolds. Ann. of Math. (2) 63 (1956), 20–63.
  61. ^ Nishida, Takaaki. A note on a theorem of Nirenberg. J. Differential Geometry 12 (1977), no. 4, 629–633
  62. ^ Nishida, Takaaki. Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation. Comm. Math. Phys. 61 (1978), no. 2, 119–148.
  63. ^ Nash, John. C1 isometric imbeddings. Ann. of Math. (2) 60 (1954), 383–396.
  64. ^ Cheng, Shiu Yuen; Yau, Shing Tung. Hypersurfaces with constant scalar curvature. Math. Ann. 225 (1977), no. 3, 195–204.
  65. ^ Rosenberg, Harold. Hypersurfaces of constant curvature in space forms. Bull. Sci. Math. 117 (1993), no. 2, 211–239.

External links edit

  • Homepage of Louis Nirenberg
  • Simons Foundation, Science Lives: Louis Nirenberg
  • Allyn Jackson. Interview with Louis Nirenberg. Notices Amer. Math. Soc. 49 (2002), no. 4, 441–449.
  • YanYan Li. The work of Louis Nirenberg. Proceedings of the International Congress of Mathematicians. Volume I, 127–137, Hindustan Book Agency, New Delhi, 2010.
  • Simon Donaldson. On the work of Louis Nirenberg. Notices Amer. Math. Soc. 58 (2011), no. 3, 469–472.
  • Tristan Rivière. Exploring the unknown: the work of Louis Nirenberg on partial differential equations. Notices Amer. Math. Soc. 63 (2016), no. 2, 120–125.
  • Recent applications of Nirenberg's classical ideas. Communicated by Christina Sormani. Notices Amer. Math. Soc. 63 (2016), no. 2, 126–134.
  • Martin Raussen and Christian Skau. Interview with Louis Nirenberg. Notices Amer. Math. Soc. 63 (2016), no. 2, 135–140.
  • (Coordinated by Robert V. Kohn and Yanyan Li.) Louis Nirenberg (1925–2020). Notices Amer. Math. Soc. 68 (2021), no. 6, 959–979.

January 01, 1970
louis, nirenberg, february, 1925, january, 2020, canadian, american, mathematician, considered, most, outstanding, mathematicians, 20th, century, nirenberg, 1975born, 1925, february, 1925hamilton, ontario, canadadiedjanuary, 2020, 2020, aged, manhattan, york, . Louis Nirenberg February 28 1925 January 26 2020 was a Canadian American mathematician considered one of the most outstanding mathematicians of the 20th century 2 3 Louis NirenbergNirenberg in 1975Born 1925 02 28 February 28 1925Hamilton Ontario CanadaDiedJanuary 26 2020 2020 01 26 aged 94 Manhattan New York U S CitizenshipCanadian and AmericanAlma materMcGill University BS 1945 New York University PhD 1950 Known forPartial differential equationsGagliardo Nirenberg interpolation inequalityGagliardo Nirenberg Sobolev inequalityBounded mean oscillation John Nirenberg space Nirenberg s conjecture 1 AwardsBocher Memorial Prize 1959 Crafoord Prize 1982 Steele Prize 1994 2014 National Medal of Science 1995 Chern Medal 2010 Abel Prize in Mathematics 2015 Scientific careerFieldsMathematicsInstitutionsNew York UniversityThesisThe determination of a closed convex surface having given line elements 1949 Doctoral advisorJames StokerDoctoral studentsWalter Craig Peter B Gilkey Djairo Guedes de Figueiredo Sergiu Klainerman YanYan Li Chang Shou Lin Wei Ming Ni Martin Schechter Gabriella TarantelloNotesCourant Institute of Mathematical Sciences Nearly all of his work was in the field of partial differential equations Many of his contributions are now regarded as fundamental to the field such as his strong maximum principle for second order parabolic partial differential equations and the Newlander Nirenberg theorem in complex geometry He is regarded as a foundational figure in the field of geometric analysis with many of his works being closely related to the study of complex analysis and differential geometry 4 Contents 1 Biography 2 Mathematical achievements 2 1 Navier Stokes equations 2 2 Nonlinear elliptic partial differential equations 2 3 Linear elliptic systems 2 4 Maximum principle and its applications 2 5 Functional inequalities 2 6 Calculus of variations 2 7 Nonlinear functional analysis 2 8 Geometric problems 2 9 Pseudo differential operators 3 Major publications 4 References 5 External linksBiography editNirenberg was born in Hamilton Ontario to Ukrainian Jewish immigrants He attended Baron Byng High School and McGill University completing his BS in both mathematics and physics in 1945 Through a summer job at the National Research Council of Canada he came to know Ernest Courant s wife Sara Paul She spoke to Courant s father the eminent mathematician Richard Courant for advice on where Nirenberg should apply to study theoretical physics Following their discussion Nirenberg was invited to enter graduate school at the Courant Institute of Mathematical Sciences at New York University In 1949 he obtained his doctorate in mathematics under the direction of James Stoker In his doctoral work he solved the Weyl problem in differential geometry which had been a well known open problem since 1916 Following his doctorate he became a professor at the Courant Institute where he remained for the rest of his career He was the advisor of 45 PhD students and published over 150 papers with a number of coauthors including notable collaborations with Henri Berestycki Haim Brezis Luis Caffarelli and Yanyan Li among many others He continued to carry out mathematical research until the age of 87 On January 26 2020 Nirenberg died at the age of 94 5 6 7 Nirenberg s work was widely recognized including the following awards and honors Bocher Memorial Prize 1959 8 Elected member of the American Academy of Arts and Sciences 1965 9 Elected member of the United States National Academy of Sciences 1969 10 Crafoord Prize 1982 11 Jeffery Williams Prize 1987 12 Elected member of the American Philosophical Society 1987 13 Steele Prize for Lifetime Achievement 1994 14 National Medal of Science 1995 15 Chern Medal 2010 16 Steele Prize for Seminal Contribution to Research 2014 with Luis Caffarelli and Robert Kohn for their article CKN82 on the Navier Stokes equations Abel Prize 2015 17 Mathematical achievements editNirenberg is especially known for his collaboration with Shmuel Agmon and Avron Douglis in which they extended the Schauder theory as previously understood for second order elliptic partial differential equations to the general setting of elliptic systems With Basilis Gidas and Wei Ming Ni he made innovative uses of the maximum principle to prove symmetry of many solutions of differential equations The study of the BMO function space was initiated by Nirenberg and Fritz John in 1961 while it was originally introduced by John in the study of elastic materials it has also been applied to games of chance known as martingales 18 His 1982 work with Luis Caffarelli and Robert Kohn made a seminal contribution to the Navier Stokes existence and smoothness in the field of mathematical fluid mechanics Other achievements include the resolution of the Minkowski problem in two dimensions the Gagliardo Nirenberg interpolation inequality the Newlander Nirenberg theorem in complex geometry and the development of pseudo differential operators with Joseph Kohn Navier Stokes equations edit The Navier Stokes equations were developed in the early 1800s to model the physics of fluid mechanics Jean Leray in a seminal achievement in the 1930s formulated an influential notion of weak solution for the equations and proved their existence 19 His work was later put into the setting of a boundary value problem by Eberhard Hopf 20 A breakthrough came with work of Vladimir Scheffer in the 1970s He showed that if a smooth solution of the Navier Stokes equations approaches a singular time then the solution can be extended continuously to the singular time away from roughly speaking a curve in space 21 Without making such a conditional assumption on smoothness he established the existence of Leray Hopf solutions which are smooth away from a two dimensional surface in spacetime 22 Such results are referred to as partial regularity Soon afterwards Luis Caffarelli Robert Kohn and Nirenberg localized and sharpened Scheffer s analysis CKN82 The key tool of Scheffer s analysis was an energy inequality providing localized integral control of solutions It is not automatically satisfied by Leray Hopf solutions but Scheffer and Caffarelli Kohn Nirenberg established existence theorems for solutions satisfying such inequalities With such a priori control as a starting point Caffarelli Kohn Nirenberg were able to prove a purely local result on smoothness away from a curve in spacetime improving Scheffer s partial regularity Similar results were later found by Michael Struwe and a simplified version of Caffarelli Kohn Nirenberg s analysis was later found by Fang Hua Lin 23 24 In 2014 the American Mathematical Society recognized Caffarelli Kohn Nirenberg s paper with the Steele Prize for Seminal Contribution to Research saying that their work was a landmark providing a source of inspiration for a generation of mathematicians The further analysis of the regularity theory of the Navier Stokes equations is as of 2021 a well known open problem Nonlinear elliptic partial differential equations edit In the 1930s Charles Morrey found the basic regularity theory of quasilinear elliptic partial differential equations for functions on two dimensional domains 25 Nirenberg as part of his Ph D thesis extended Morrey s results to the setting of fully nonlinear elliptic equations N53a The works of Morrey and Nirenberg made extensive use of two dimensionality and the understanding of elliptic equations with higher dimensional domains was an outstanding open problem The Monge Ampere equation in the form of prescribing the determinant of the hessian of a function is one of the standard examples of a fully nonlinear elliptic equation In an invited lecture at the 1974 International Congress of Mathematicians Nirenberg announced results obtained with Eugenio Calabi on the boundary value problem for the Monge Ampere equation based upon boundary regularity estimates and a method of continuity 26 However they soon realized their proofs to be incomplete 26 In 1977 Shiu Yuen Cheng and Shing Tung Yau resolved the existence and interior regularity for the Monge Ampere equation showing in particular that if the determinant of the hessian of a function is smooth then the function itself must be smooth as well 27 Their work was based upon the relation via the Legendre transform to the Minkowski problem which they had previously resolved by differential geometric estimates 28 In particular their work did not make use of boundary regularity and their results left such questions unresolved In collaboration with Luis Caffarelli and Joel Spruck Nirenberg resolved such questions directly establishing boundary regularity and using it to build a direct approach to the Monge Ampere equation based upon the method of continuity CNS84 Calabi and Nirenberg had successfully demonstrated uniform control of the first two derivatives the key for the method of continuity is the more powerful uniform Holder continuity of the second derivatives Caffarelli Nirenberg and Spruck established a delicate version of this along the boundary 29 which they were able to establish as sufficient by using Calabi s third derivative estimates in the interior 30 With Joseph Kohn they found analogous results in the setting of the complex Monge Ampere equation C 85 In such general situations the Evans Krylov theory 29 is a more flexible tool than the computation based calculations of Calabi Caffarelli Nirenberg and Spruck were able to extend their methods to more general classes of fully nonlinear elliptic partial differential equations in which one studies functions for which certain relations between the hessian s eigenvalues are prescribed CNS85 As a particular case of their new class of equations they were able to partially resolve the boundary value problem for special Lagrangians Linear elliptic systems edit Nirenberg s most renowned work from the 1950s deals with elliptic regularity With Avron Douglis Nirenberg extended the Schauder estimates as discovered in the 1930s in the context of second order elliptic equations to general elliptic systems of arbitrary order DN55 In collaboration with Shmuel Agmon and Douglis Nirenberg proved boundary regularity for elliptic equations of arbitrary order ADN59 They later extended their results to elliptic systems of arbitrary order ADN64 With Morrey Nirenberg proved that solutions of elliptic systems with analytic coefficients are themselves analytic extending to the boundary earlier known work MN57 These contributions to elliptic regularity are now considered as part of a standard package of information and are covered in many textbooks The Douglis Nirenberg and Agmon Douglis Nirenberg estimates in particular are among the most widely used tools in elliptic partial differential equations 31 With Yanyan Li and motivated by composite materials in elasticity theory Nirenberg studied linear elliptic systems in which the coefficients are Holder continuous in the interior but possibly discontinuous on the boundary Their result is that the gradient of the solution is Holder continuous with a L estimate for the gradient which is independent of the distance from the boundary LN03 Maximum principle and its applications edit In the case of harmonic functions the maximum principle was known in the 1800s and was used by Carl Friedrich Gauss 32 33 In the early 1900s complicated extensions to general second order elliptic partial differential equations were found by Sergei Bernstein Leon Lichtenstein and Emile Picard it was not until the 1920s that the simple modern proof was found by Eberhard Hopf 34 In one of his earliest works Nirenberg adapted Hopf s proof to second order parabolic partial differential equations thereby establishing the strong maximum principle in that context N53b As in the earlier work such a result had various uniqueness and comparison theorems as corollaries Nirenberg s work is now regarded as one of the foundations of the field of parabolic partial differential equations and is ubiquitous across the standard textbooks 35 36 37 38 39 40 In the 1950s A D Alexandrov introduced an elegant moving plane reflection method which he used as the context for applying the maximum principle to characterize the standard sphere as the only closed hypersurface of Euclidean space with constant mean curvature In 1971 James Serrin utilized Alexandrov s technique to prove that highly symmetric solutions of certain second order elliptic partial differential equations must be supported on symmetric domains Nirenberg realized that Serrin s work could be reformulated so as to prove that solutions of second order elliptic partial differential equations inherit symmetries of their domain and of the equation itself Such results do not hold automatically and it is nontrivial to identify which special features of a given problem are relevant For example there are many harmonic functions on Euclidean space which fail to be rotationally symmetric despite the rotational symmetry of the Laplacian and of Euclidean space Nirenberg s first results on this problem were obtained in collaboration with Basilis Gidas and Wei Ming Ni They developed a precise form of Alexandrov and Serrin s technique applicable even to fully nonlinear elliptic and parabolic equations GNN79 In a later work they developed a version of the Hopf lemma applicable on unbounded domains thereby improving their work in the case of equations on such domains GNN81 Their main applications deal with rotational symmetry Due to such results in many cases of geometric or physical interest it is sufficient to study ordinary differential equations rather than partial differential equations Later with Henri Berestycki Nirenberg used the Alexandrov Bakelman Pucci estimate 29 to improve and modify the methods of Gidas Ni Nirenberg significantly reducing the need to assume regularity of the domain BN91a In an important result with Srinivasa Varadhan Berestycki and Nirenberg continued the study of domains with no assumed regularity For linear operators they related the validity of the maximum principle to positivity of a first eigenvalue and existence of a first eigenfunction BNV94 With Luis Caffarelli Berestycki and Nirenberg applied their results to symmetry of functions on cylindrical domains BCN96 They obtained in particular a partial resolution of a well known conjecture of Ennio De Giorgi on translational symmetry which was later fully resolved in Ovidiu Savin s Ph D thesis BCN97b 41 42 They further applied their method to obtain qualitative phenomena on general unbounded domains extending earlier works of Maria Esteban and Pierre Louis Lions BCN97a Functional inequalities edit Nirenberg and Emilio Gagliardo independently proved fundamental inequalities for Sobolev spaces now known as the Gagliardo Nirenberg Sobolev inequality and the Gagliardo Nirenberg interpolation inequalities N59 They are used ubiquitously throughout the literature on partial differential equations as such it has been of great interest to extend and adapt them to various situations Nirenberg himself would later clarify the possible exponents which can appear in the interpolation inequality N66 With Luis Caffarelli and Robert Kohn Nirenberg would establish corresponding inequalities for certain weighted norms CKN84 Caffarelli Kohn and Nirenberg s norms were later investigated more fully in notable work by Florin Catrina and Zhi Qiang Wang 43 Immediately following Fritz John s introduction of the bounded mean oscillation BMO function space in the theory of elasticity he and Nirenberg gave a further study of the space proving in particular the John Nirenberg inequality which constrains the size of the set on which a BMO function is far from its average value JN61 Their work which is an application of the Calderon Zygmund decomposition has become a part of the standard mathematical literature Expositions are contained in standard textbooks on probability 44 complex analysis 45 harmonic analysis 46 Fourier analysis 47 and partial differential equations 29 Among other applications it is particularly fundamental to Jurgen Moser s Harnack inequality and subsequent work 48 49 29 The John Nirenberg inequality and the more general foundations of the BMO theory were worked out by Nirenberg and Haim Brezis in the context of maps between Riemannian manifolds BN95 Among other results they were able to establish that smooth maps which are close in BMO norm have the same topological degree and hence that degree can be meaningfully defined for mappings of vanishing mean oscillation VMO Calculus of variations edit In the setting of topological vector spaces Ky Fan developed a minimax theorem with applications in game theory 50 51 With Haim Brezis and Guido Stampacchia Nirenberg derived results extending both Fan s theory and Stampacchia s generalization of the Lax Milgram theorem BNS72 52 Their work has applications to the subject of variational inequalities 53 By adapting the Dirichlet energy it is standard to recognize solutions of certain wave equations as critical points of functionals With Brezis and Jean Michel Coron Nirenberg found a novel functional whose critical points can be directly used to construct solutions of wave equations BCN80 They were able to apply the mountain pass theorem to their new functional thereby establishing the existence of periodic solutions of certain wave equations extending a result of Paul Rabinowitz 54 Part of their work involved small extensions of the standard mountain pass theorem and Palais Smale condition which have become standard in textbooks 55 56 57 In 1991 Brezis and Nirenberg showed how Ekeland s variational principle could be applied to extend the mountain pass theorem with the effect that almost critical points can be found without requiring the Palais Smale condition BN91b 57 A fundamental contribution of Brezis and Nirenberg to critical point theory dealt with local minimizers BN93 In principle the choice of function space is highly relevant and a function could minimize among smooth functions without minimizing among the broader class of Sobolev functions Making use of an earlier regularity result of Brezis and Tosio Kato Brezis and Nirenberg ruled out such phenomena for a certain class of Dirichlet type functionals 58 Their work was later extended by Jesus Garcia Azorero Juan Manfredi and Ireneo Peral 59 In one of Nirenberg s most widely cited papers he and Brezis studied the Dirichlet problem for Yamabe type equations on Euclidean spaces following part of Thierry Aubin s work on the Yamabe problem BN83 With Berestycki and Italo Capuzzo Dolcetta Nirenberg studied superlinear equations of Yamabe type giving various existence and non existence results BCN94 Nonlinear functional analysis edit Agmon and Nirenberg made an extensive study of ordinary differential equations in Banach spaces relating asymptotic representations and the behavior at infinity of solutions to d u d t A u 0 displaystyle frac du dt Au 0 nbsp to the spectral properties of the operator A Applications include the study of rather general parabolic and elliptic parabolic problems AN63 Brezis and Nirenberg gave a study of the perturbation theory of nonlinear perturbations of noninvertible transformations between Hilbert spaces applications include existence results for periodic solutions of some semilinear wave equations BN78a BN78b In John Nash s work on the isometric embedding problem the key step is a small perturbation result highly reminiscent of an implicit function theorem his proof used a novel combination of Newton s method in an infinitesimal form with smoothing operators 60 Nirenberg was one of many mathematicians to put Nash s ideas into systematic and abstract frameworks referred to as Nash Moser theorems Nirenberg s formulation is particularly simple isolating the basic analytic ideas underlying the analysis of most Nash Moser iteration schemes N72 Within a similar framework he proved an abstract form of the Cauchy Kowalevski theorem as a particular case of a theorem on solvability of ordinary differential equations in families of Banach spaces N72 His work was later simplified by Takaaki Nishida and used in an analysis of the Boltzmann equation 61 62 Geometric problems edit Making use of his work on fully nonlinear elliptic equations N53a Nirenberg s Ph D thesis provided a resolution of the Weyl problem and Minkowski problem in the field of differential geometry N53c The former asks for the existence of isometric embeddings of positively curved Riemannian metrics on the two dimensional sphere into three dimensional Euclidean space while the latter asks for closed surfaces in three dimensional Euclidean space for which the Gauss map prescribes the Gaussian curvature The key is that the Darboux equation from surface theory is of Monge Ampere type so that Nirenberg s regularity theory becomes useful in the method of continuity John Nash s well known isometric embedding theorems established soon afterwards have no apparent relation to the Weyl problem which deals simultaneously with high regularity embeddings and low codimension 63 60 Nirenberg s work on the Minkowski problem was extended to Riemannian settings by Aleksei Pogorelov In higher dimensions the Minkowski problem was resolved by Shiu Yuen Cheng and Shing Tung Yau 28 Other approaches to the Minkowski problem have developed from Caffarelli Nirenberg and Spruck s fundamental contributions to the theory of nonlinear elliptic equations CNS85 In one of his very few articles not centered on analysis Nirenberg and Philip Hartman characterized the cylinders in Euclidean space as the only complete hypersurfaces which are intrinsically flat HN59 This can also be viewed as resolving a question on the isometric embedding of flat manifolds as hypersurfaces Such questions and natural generalizations were later taken up by Cheng Yau and Harold Rosenberg among others 64 65 Answering a question posed to Nirenberg by Shiing Shen Chern and Andre Weil Nirenberg and his doctoral student August Newlander proved what is now known as the Newlander Nirenberg theorem which provides the precise algebraic condition under which an almost complex structure arises from a holomorphic coordinate atlas NN57 The Newlander Nirenberg theorem is now considered as a foundational result in complex geometry although the result itself is far better known than the proof which is not usually covered in introductory texts as it relies on advanced methods in partial differential equations Nirenberg and Joseph Kohn following earlier work by Kohn studied the Neumann problem on pseudoconvex domains and demonstrated the relation of the regularity theory to the existence of subelliptic estimates for the operator KN65b The classical Poincare disk model assigns the metric of hyperbolic space to the unit ball Nirenberg and Charles Loewner studied the more general means of naturally assigning a complete Riemannian metric to bounded open subsets of Euclidean space LN74 Geometric calculations show that solutions of certain semilinear Yamabe type equations can be used to define metrics of constant scalar curvature and that the metric is complete if the solution diverges to infinity near the boundary Loewner and Nirenberg established existence of such solutions on certain domains Similarly they studied a certain Monge Ampere equation with the property that for any negative solution extending continuously to zero at the boundary one can define a complete Riemannian metric via the hessian These metrics have the special property of projective invariance so that projective transformation from one given domain to another becomes an isometry of the corresponding metrics Pseudo differential operators edit Joseph Kohn and Nirenberg introduced the notion of pseudo differential operators KN65a Nirenberg and Francois Treves investigated the famous Lewy s example for a non solvable linear PDE of second order and discovered the conditions under which it is solvable in the context of both partial differential operators and pseudo differential operators NT63 NT70 Their introduction of local solvability conditions with analytic coefficients has become a focus for researchers such as R Beals C Fefferman R D Moyer Lars Hormander and Nils Dencker who solved the pseudo differential condition for Lewy s equation This opened up further doors into the local solvability of linear partial differential equations Major publications editBooks and surveys N73 Nirenberg Louis 1973 Lectures on linear partial differential equations Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics Vol 17 Providence RI American Mathematical Society doi 10 1090 cbms 017 ISBN 978 0 8218 1667 7 MR 0450755 Zbl 0267 35001 N81 Nirenberg L 1981 Variational and topological methods in nonlinear problems Bulletin of the American Mathematical Society New Series 4 3 267 302 doi 10 1090 S0273 0979 1981 14888 6 MR 0609039 Zbl 0468 47040 N94 Nirenberg Louis 1994 Partial differential equations in the first half of the century In Pier Jean Paul ed Development of Mathematics 1900 1950 Symposium held in Luxembourg June 1992 Basel Birkhauser pp 479 515 ISBN 3 7643 2821 5 MR 1298641 Zbl 0807 01017 N01 Nirenberg Louis 2001 Topics in nonlinear functional analysis Courant Lecture Notes in Mathematics Vol 6 Chapter 6 by E Zehnder Notes by R A Artino Revised reprint of the 1974 original ed New York New York University Courant Institute of Mathematical Sciences doi 10 1090 cln 006 ISBN 0 8218 2819 3 MR 0488102 Zbl 0286 47037 N18 Nirenberg Louis 2018 Lectures on differential equations and differential geometry Classical Topics in Mathematics Vol 7 With a preface by Shiu Yuen Cheng and Lizhen Ji Beijing Higher Education Press ISBN 978 7 04 050302 9 MR 3889014 Zbl 1465 35001 Articles N53a Nirenberg Louis 1953 On nonlinear elliptic partial differential equations and Holder continuity Communications on Pure and Applied Mathematics 6 1 103 156 doi 10 1002 cpa 3160060105 MR 0064986 Zbl 0050 09801 Erratum doi 10 1002 cpa 3160060304 N53b Nirenberg Louis 1953 A strong maximum principle for parabolic equations Communications on Pure and Applied Mathematics 6 2 167 177 doi 10 1002 cpa 3160060202 MR 0055544 Zbl 0050 09601 N53c Nirenberg Louis 1953 The Weyl and Minkowski problems in differential geometry in the large Communications on Pure and Applied Mathematics 6 3 337 394 doi 10 1002 cpa 3160060303 MR 0058265 Zbl 0051 12402 DN55 Douglis Avron Nirenberg Louis 1955 Interior estimates for elliptic systems of partial differential equations Communications on Pure and Applied Mathematics 8 4 503 538 doi 10 1002 cpa 3160080406 MR 0075417 Zbl 0066 08002 N55 Nirenberg Louis 1955 Remarks on strongly elliptic partial differential equations Communications on Pure and Applied Mathematics 8 4 649 675 doi 10 1002 cpa 3160080414 MR 0075415 Zbl 0067 07602 MN57 Morrey C B Jr Nirenberg L 1957 On the analyticity of the solutions of linear elliptic systems of partial differential equations Communications on Pure and Applied Mathematics 10 2 271 290 doi 10 1002 cpa 3160100204 MR 0089334 Zbl 0082 09402 NN57 Newlander A Nirenberg L 1957 Complex analytic coordinates in almost complex manifolds Annals of Mathematics Second Series 65 3 391 404 doi 10 2307 1970051 JSTOR 1970051 MR 0088770 Zbl 0079 16102 ADN59 Agmon S Douglis A Nirenberg L 1959 Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I Communications on Pure and Applied Mathematics 12 4 623 727 doi 10 1002 cpa 3160120405 MR 0125307 Zbl 0093 10401 HN59 Hartman Philip Nirenberg Louis 1959 On spherical image maps whose Jacobians do not change sign American Journal of Mathematics 81 4 901 920 doi 10 2307 2372995 JSTOR 2372995 MR 0126812 Zbl 0094 16303 N59 Nirenberg L 1959 On elliptic partial differential equations Annali della Scuola Normale Superiore di Pisa Classe di Scienze Serie III 13 2 115 162 MR 0109940 Zbl 0088 07601 JN61 John F Nirenberg L 1961 On functions of bounded mean oscillation Communications on Pure and Applied Mathematics 14 3 415 426 doi 10 1002 cpa 3160140317 MR 0131498 Zbl 0102 04302 AN63 Agmon S Nirenberg L 1963 Properties of solutions of ordinary differential equations in Banach space Communications on Pure and Applied Mathematics 16 2 121 239 doi 10 1002 cpa 3160160204 MR 0155203 Zbl 0117 10001 NT63 Nirenberg L Treves F 1963 Solvability of a first order linear partial differential equation Communications on Pure and Applied Mathematics 16 3 331 351 doi 10 1002 cpa 3160160308 MR 0163045 Zbl 0117 06104 ADN64 Agmon S Douglis A Nirenberg L 1964 Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II Communications on Pure and Applied Mathematics 17 1 35 92 doi 10 1002 cpa 3160170104 MR 0162050 Zbl 0123 28706 KN65a Kohn J J Nirenberg L 1965 An algebra of pseudo differential operators Communications on Pure and Applied Mathematics 18 1 2 269 305 doi 10 1002 cpa 3160180121 MR 0176362 Zbl 0171 35101 KN65b Kohn J J Nirenberg L 1965 Non coercive boundary value problems Communications on Pure and Applied Mathematics 18 3 443 492 doi 10 1002 cpa 3160180305 MR 0181815 Zbl 0125 33302 N66 Nirenberg L 1966 An extended interpolation inequality Annali della Scuola Normale Superiore di Pisa Classe di Scienze Serie III 20 4 733 737 MR 0208360 Zbl 0163 29905 KN67 Kohn J J Nirenberg L 1967 Degenerate elliptic parabolic equations of second order Communications on Pure and Applied Mathematics 20 4 797 872 doi 10 1002 cpa 3160200410 MR 0234118 Zbl 0153 14503 NT70 Nirenberg Louis Treves Francois 1970 On local solvability of linear partial differential equations I Necessary conditions Communications on Pure and Applied Mathematics 23 1 1 38 doi 10 1002 cpa 3160230102 MR 0264470 Zbl 0191 39103 BNS72 Brezis H Nirenberg L Stampacchia G 1972 A remark on Ky Fan s minimax principle Bollettino della Unione Matematica Italiana Series IV 6 293 300 MR 0324498 Zbl 0264 49013 N72 Nirenberg L 1972 An abstract form of the nonlinear Cauchy Kowalewski theorem Journal of Differential Geometry 6 4 561 576 doi 10 4310 jdg 1214430643 MR 0322321 Zbl 0257 35001 NW73 Nirenberg Louis Walker Homer F 1973 The null spaces of elliptic partial differential operators in Rn Journal of Mathematical Analysis and Applications 42 2 271 301 doi 10 1016 0022 247X 73 90138 8 MR 0320821 Zbl 0272 35029 LN74 Loewner Charles Nirenberg Louis 1974 Partial differential equations invariant under conformal or projective transformations In Ahlfors Lars V Kra Irwin Maskit Bernard Nirenberg Louis eds Contributions to Analysis A Collection of Papers Dedicated to Lipman Bers New York Academic Press pp 245 272 doi 10 1016 B978 0 12 044850 0 50027 7 MR 0358078 Zbl 0298 35018 KN77 Kinderlehrer D Nirenberg L 1977 Regularity in free boundary problems Annali della Scuola Normale Superiore di Pisa Classe di Scienze Serie IV 4 2 373 391 MR 0440187 Zbl 0352 35023 BN78a Brezis H Nirenberg L 1978 Characterizations of the ranges of some nonlinear operators and applications to boundary value problems Annali della Scuola Normale Superiore di Pisa Classe di Scienze Serie IV 5 2 225 326 MR 0513090 Zbl 0386 47035 BN78b Brezis H Nirenberg L 1978 Forced vibrations for a nonlinear wave equation Communications on Pure and Applied Mathematics 31 1 1 30 doi 10 1002 cpa 3160310102 MR 0470377 Zbl 0378 35040 GNN79 Gidas B Ni Wei Ming Nirenberg L 1979 Symmetry and related properties via the maximum principle Communications in Mathematical Physics 68 3 209 243 Bibcode 1979CMaPh 68 209G doi 10 1007 BF01221125 MR 0544879 S2CID 56251822 Zbl 0425 35020 BCN80 Brezis Haim Coron Jean Michel Nirenberg Louis 1980 Free vibrations for a nonlinear wave equation and a theorem of P Rabinowitz Communications on Pure and Applied Mathematics 33 5 667 684 doi 10 1002 cpa 3160330507 MR 0586417 Zbl 0484 35057 GNN81 Gidas B Ni Wei Ming Nirenberg L 1981 Symmetry of positive solutions of nonlinear elliptic equations in Rn In Nachbin Leopoldo ed Mathematical Analysis and Applications Part A Essays dedicated to Laurent Schwartz on the occasion of his 65th birthday Advances in Mathematics Supplementary Studies Vol 7a New York London Academic Press pp 369 402 ISBN 0 12 512801 0 MR 0634248 Zbl 0469 35052 CKN82 Caffarelli L Kohn R Nirenberg L 1982 Partial regularity of suitable weak solutions of the Navier Stokes equations Communications on Pure and Applied Mathematics 35 6 771 831 Bibcode 1982CPAM 35 771C doi 10 1002 cpa 3160350604 MR 0673830 Zbl 0509 35067 BN83 Brezis Haim Nirenberg Louis 1983 Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents Communications on Pure and Applied Mathematics 36 4 437 477 doi 10 1002 cpa 3160360405 MR 0709644 Zbl 0541 35029 CKN84 Caffarelli L Kohn R Nirenberg L 1984 First order interpolation inequalities with weights Compositio Mathematica 53 3 259 275 MR 0768824 Zbl 0563 46024 CNS84 Caffarelli L Nirenberg L Spruck J 1984 The Dirichlet problem for nonlinear second order elliptic equations I Monge Ampere equation Communications on Pure and Applied Mathematics 37 3 369 402 doi 10 1002 cpa 3160370306 MR 0739925 Zbl 0598 35047 Erratum doi 10 1002 cpa 3160400508 C 85 Caffarelli L Kohn J J Nirenberg L Spruck J 1985 The Dirichlet problem for nonlinear second order elliptic equations II Complex Monge Ampere and uniformly elliptic equations Communications on Pure and Applied Mathematics 38 2 209 252 doi 10 1002 cpa 3160380206 MR 0780073 Zbl 0598 35048 CNS85 Caffarelli L Nirenberg L Spruck J 1985 The Dirichlet problem for nonlinear second order elliptic equations III Functions of the eigenvalues of the Hessian Acta Mathematica 155 3 4 261 301 doi 10 1007 BF02392544 MR 0806416 Zbl 0654 35031 BN91a Berestycki H Nirenberg L 1991 On the method of moving planes and the sliding method Boletim da Sociedade Brasileira de Matematica New Series 22 1 1 37 doi 10 1007 BF01244896 MR 1159383 S2CID 119878360 Zbl 0784 35025 BN91b Brezis Haim Nirenberg Louis 1991 Remarks on finding critical points Communications on Pure and Applied Mathematics 44 8 9 939 963 doi 10 1002 cpa 3160440808 MR 1127041 Zbl 0751 58006 BN92 Berestycki Henri Nirenberg Louis 1992 Travelling fronts in cylinders Annales de l Institut Henri Poincare C 9 5 497 572 Bibcode 1992AIHPC 9 497B doi 10 1016 S0294 1449 16 30229 3 MR 1191008 Zbl 0799 35073 BN93 Brezis Haim Nirenberg Louis 1993 H1 versus C1 local minimizers Comptes Rendus de l Academie des Sciences Serie I 317 5 465 472 MR 1239032 Zbl 0803 35029 BCN94 Berestycki H Capuzzo Dolcetta I Nirenberg L 1994 Superlinear indefinite elliptic problems and nonlinear Liouville theorems Topological Methods in Nonlinear Analysis 4 1 59 78 doi 10 12775 TMNA 1994 023 MR 1321809 Zbl 0816 35030 BNV94 Berestycki H Nirenberg L Varadhan S R S 1994 The principal eigenvalue and maximum principle for second order elliptic operators in general domains Communications on Pure and Applied Mathematics 47 1 47 92 doi 10 1002 cpa 3160470105 MR 1258192 Zbl 0806 35129 BCN95 Berestycki Henri Capuzzo Dolcetta Italo Nirenberg Louis 1995 Variational methods for indefinite superlinear homogeneous elliptic problems Nonlinear Differential Equations and Applications 2 4 553 572 doi 10 1007 BF01210623 MR 1356874 S2CID 119668931 Zbl 0840 35035 BN95 Brezis H Nirenberg L 1995 Degree theory and BMO I Compact manifolds without boundaries Selecta Mathematica New Series 1 2 197 263 doi 10 1007 BF01671566 MR 1354598 S2CID 195270732 Zbl 0852 58010 BCN96 Berestycki H Caffarelli L A Nirenberg L 1996 Inequalities for second order elliptic equations with applications to unbounded domains I Duke Mathematical Journal 81 2 467 494 doi 10 1215 S0012 7094 96 08117 X MR 1395408 Zbl 0860 35004 BCN97a Berestycki H Caffarelli L A Nirenberg L 1997 Monotonicity for elliptic equations in unbounded Lipschitz domains Communications on Pure and Applied Mathematics 50 11 1089 1111 doi 10 1002 SICI 1097 0312 199711 50 11 lt 1089 AID CPA2 gt 3 0 CO 2 6 MR 1470317 Zbl 0906 35035 BCN97b Berestycki Henri Caffarelli Luis Nirenberg Louis 1997 Further qualitative properties for elliptic equations in unbounded domains Annali della Scuola Normale Superiore di Pisa Classe di Scienze Serie IV 25 1 2 69 94 MR 1655510 Zbl 1079 35513 LN03 Li Yanyan Nirenberg Louis 2003 Estimates for elliptic systems from composite material Communications on Pure and Applied Mathematics 56 7 892 925 doi 10 1002 cpa 10079 MR 1990481 S2CID 122040650 Zbl 1125 35339 References edit Lawson H Blaine Jr April 21 2012 Reflections on the Early Mathematical Life of Bob Osserman PDF Allyn Jackson March 2002 Interview with Louis Nirenberg PDF Notices of the AMS 49 4 441 449 Archived from the original PDF on March 3 2016 Retrieved March 26 2015 Caffarelli Luis A Li YanYan Preface Dedicated to Louis Nirenberg on the occasion of his 85th birthday Part I Discrete Contin Dyn Syst 28 2010 no 2 i ii doi 10 3934 dcds 2010 28 2i Yau Shing Tung Perspectives on geometric analysis Surveys in differential geometry Vol X 275 379 Surv Differ Geom 10 Int Press Somerville MA 2006 Morto il grande matematico Louis Nirenberg in Italian Chang Kenneth January 31 2020 Louis Nirenberg One of the Great Mathematicians Dies at 94 New York Times Retrieved February 19 2020 Shields Brit Barany Michael J February 17 2020 Louis Nirenberg 1925 2020 Nature Retrieved February 19 2020 Browse Prizes and Awards American Mathematical Society Retrieved August 12 2022 Louis Nirenberg American Academy of Arts amp Sciences Retrieved May 5 2022 Louis Nirenberg www nasonline org Retrieved May 5 2022 The Crafoord Prize 1982 The Crafoord Prize May 25 1982 Jeffery Williams Prize CMS SMC Retrieved August 12 2022 APS Member History search amphilsoc org Retrieved May 5 2022 1994 Steele Prizes Notices Amer Math Soc 41 1994 no 8 905 912 Louis Nirenberg receives National Medal of Science With contributions by Luis Caffarelli and Joseph J Kohn Notices Amer Math Soc 43 1996 no 10 1111 1116 2010 Chern Medal awarded Notices Amer Math Soc 57 2010 no 11 1472 1474 2015 John F Nash and Louis Nirenberg The Norwegian Academy of Science and Letters Retrieved August 12 2022 John F Nash Jr and Louis Nirenberg share the Abel Prize The Abel Prize March 25 2015 Retrieved March 26 2015 Leray Jean Sur le mouvement d un liquide visqueux emplissant l espace Acta Math 63 1934 no 1 193 248 Hopf Eberhard Uber die Anfangswertaufgabe fur die hydrodynamischen Grundgleichungen Math Nachr 4 1951 213 231 Scheffer Vladimir Partial regularity of solutions to the Navier Stokes equations Pacific J Math 66 1976 no 2 535 552 Scheffer Vladimir Hausdorff measure and the Navier Stokes equations Comm Math Phys 55 1977 no 2 97 112 Struwe Michael On partial regularity results for the Navier Stokes equations Comm Pure Appl Math 41 1988 no 4 437 458 Lin Fanghua A new proof of the Caffarelli Kohn Nirenberg theorem Comm Pure Appl Math 51 1998 no 3 241 257 Morrey Charles B Jr On the solutions of quasi linear elliptic partial differential equations Trans Amer Math Soc 43 1938 no 1 126 166 a b See the second page of CNS84 Cheng Shiu Yuen Yau Shing Tung On the regularity of the Monge Ampere equation det 2u xi xj F x u Comm Pure Appl Math 30 1977 no 1 41 68 a b Cheng Shiu Yuen Yau Shing Tung On the regularity of the solution of the n dimensional Minkowski problem Comm Pure Appl Math 29 1976 no 5 495 516 a b c d e Gilbarg David Trudinger Neil S Elliptic partial differential equations of second order Reprint of the 1998 edition Classics in Mathematics Springer Verlag Berlin 2001 xiv 517 pp ISBN 3 540 41160 7 Calabi Eugenio Improper affine hyperspheres of convex type and a generalization of a theorem by K Jorgens Michigan Math J 5 1958 105 126 Morrey Charles B Jr Multiple integrals in the calculus of variations Die Grundlehren der mathematischen Wissenschaften Band 130 Springer Verlag New York Inc New York 1966 ix 506 pp The historical comments and references are taken from James Serrin s commentary on page 9 of Selected Works of Eberhard Hopf with Commentaries American Mathematical Society Providence RI 2002 xxiv 386 pp Gauss C F Allgemeine Theorie des Erdmagnetismus Resultate aus den Beobachtungen des Magnetischen Vereins im Jahre 1838 Hopf Eberhard Elementare Bemerkungen uber die Losungen partieller Differentialgleichngen zweiter Ordnung vom elliptischen Typus 1927 Evans Lawrence C Partial differential equations Second edition Graduate Studies in Mathematics 19 American Mathematical Society Providence RI 2010 xxii 749 pp Friedman Avner Partial differential equations of parabolic type Prentice Hall Inc Englewood Cliffs N J 1964 xiv 347 pp Ladyzenskaja O A Solonnikov V A Uralʹceva N N Linear and quasilinear equations of parabolic type Translations of Mathematical Monographs Vol 23 American Mathematical Society Providence R I 1968 xi 648 pp Lieberman Gary M Second order parabolic differential equations World Scientific Publishing Co Inc River Edge NJ 1996 xii 439 pp Protter Murray H Weinberger Hans F Maximum principles in differential equations Corrected reprint of the 1967 original Springer Verlag New York 1984 x 261 pp Smoller Joel Shock waves and reaction diffusion equations Second edition Grundlehren der mathematischen Wissenschaften 258 Springer Verlag New York 1994 xxiv 632 pp Savin Ovidiu Regularity of flat level sets in phase transitions Ann of Math 2 169 2009 no 1 41 78 del Pino Manuel Kowalczyk Michal Wei Juncheng On De Giorgi s conjecture in dimension N 9 Ann of Math 2 174 2011 no 3 1485 1569 Catrina Florin Wang Zhi Qiang On the Caffarelli Kohn Nirenberg inequalities sharp constants existence and nonexistence and symmetry of extremal functions Comm Pure Appl Math 54 2001 no 2 229 258 Revuz Daniel Yor Marc Continuous martingales and Brownian motion Third edition Grundlehren der mathematischen Wissenschaften 293 Springer Verlag Berlin 1999 xiv 602 pp ISBN 3 540 64325 7 Garnett John B Bounded analytic functions Revised first edition Graduate Texts in Mathematics 236 Springer New York 2007 xiv 459 pp ISBN 978 0 387 33621 3 0 387 33621 4 Garcia Cuerva Jose Rubio de Francia Jose L Weighted norm inequalities and related topics North Holland Mathematics Studies 116 Notas de Matematica 104 North Holland Publishing Co Amsterdam 1985 x 604 pp ISBN 0 444 87804 1 Grafakos Loukas Modern Fourier analysis Third edition Graduate Texts in Mathematics 250 Springer New York 2014 xvi 624 pp ISBN 978 1 4939 1229 2 978 1 4939 1230 8 Moser Jurgen On Harnack s theorem for elliptic differential equations Comm Pure Appl Math 14 1961 577 591 Moser Jurgen A Harnack inequality for parabolic differential equations Comm Pure Appl Math 17 1964 101 134 Fan Ky A generalization of Tychonoff s fixed point theorem Math Ann 142 1960 305 310 Fan Ky A minimax inequality and applications Inequalities III Proc Third Sympos Univ California Los Angeles Calif 1969 dedicated to the memory of Theodore S Motzkin pp 103 113 Academic Press New York 1972 Stampacchia Guido Formes bilineaires coercitives sur les ensembles convexes C R Acad Sci Paris 258 1964 4413 4416 Aubin Jean Pierre Ekeland Ivar Applied nonlinear analysis Reprint of the 1984 original Dover Publications Inc Mineola NY 2006 x 518 pp Rabinowitz Paul H Free vibrations for a semilinear wave equation Comm Pure Appl Math 31 1978 no 1 31 68 Mawhin Jean Willem Michel Critical point theory and Hamiltonian systems Applied Mathematical Sciences 74 Springer Verlag New York 1989 xiv 277 pp Struwe Michael Variational methods Applications to nonlinear partial differential equations and Hamiltonian systems Fourth edition Ergebnisse der Mathematik und ihrer Grenzgebiete 3 Folge A Series of Modern Surveys in Mathematics 34 Springer Verlag Berlin 2008 xx 302 pp a b Willem Michel Minimax theorems Progress in Nonlinear Differential Equations and their Applications 24 Birkhauser Boston Inc Boston MA 1996 x 162 pp Brezis Haim Kato Tosio Remarks on the Schrodinger operator with singular complex potentials J Math Pures Appl 9 58 1979 no 2 137 151 Garcia Azorero J P Peral Alonso I Manfredi Juan J Sobolev versus Holder local minimizers and global multiplicity for some quasilinear elliptic equations Commun Contemp Math 2 2000 no 3 385 404 a b Nash John The imbedding problem for Riemannian manifolds Ann of Math 2 63 1956 20 63 Nishida Takaaki A note on a theorem of Nirenberg J Differential Geometry 12 1977 no 4 629 633 Nishida Takaaki Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation Comm Math Phys 61 1978 no 2 119 148 Nash John C1 isometric imbeddings Ann of Math 2 60 1954 383 396 Cheng Shiu Yuen Yau Shing Tung Hypersurfaces with constant scalar curvature Math Ann 225 1977 no 3 195 204 Rosenberg Harold Hypersurfaces of constant curvature in space forms Bull Sci Math 117 1993 no 2 211 239 External links editHomepage of Louis Nirenberg Simons Foundation Science Lives Louis Nirenberg Allyn Jackson Interview with Louis Nirenberg Notices Amer Math Soc 49 2002 no 4 441 449 YanYan Li The work of Louis Nirenberg Proceedings of the International Congress of Mathematicians Volume I 127 137 Hindustan Book Agency New Delhi 2010 Simon Donaldson On the work of Louis Nirenberg Notices Amer Math Soc 58 2011 no 3 469 472 Tristan Riviere Exploring the unknown the work of Louis Nirenberg on partial differential equations Notices Amer Math Soc 63 2016 no 2 120 125 Recent applications of Nirenberg s classical ideas Communicated by Christina Sormani Notices Amer Math Soc 63 2016 no 2 126 134 Martin Raussen and Christian Skau Interview with Louis Nirenberg Notices Amer Math Soc 63 2016 no 2 135 140 Coordinated by Robert V Kohn and Yanyan Li Louis Nirenberg 1925 2020 Notices Amer Math Soc 68 2021 no 6 959 979 Retrieved from https en wikipedia org w index php title Louis Nirenberg amp oldid 1213839651, wikipedia, wiki, book, books, library,

article

, read, download, free, free download, mp3, video, mp4, 3gp, jpg, jpeg, gif, png, picture, music, song, movie, book, game, games.