Louis Nirenberg: 2015 Abel Prize for his contributions to ... · "Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré L. Nirenberg's main

Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs

Abel Prize 2015 to the American mathematicians John F. Nash, Jr. and Louis Nirenberg ''for striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis''

Xavier CabréICREA Research Professor at the UPC

06/05/2015 FME-UPC

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré

L. Nirenberg: B.Sc. from McGill University (Canada), 1945.

Louis Nirenberg:

● Born Feb. 28, 1925 in Hamilton, Ontario, Canada, in a Jewish family

● Master and Graduate School at New York University

● Ph.D. 1949 under the direction of James Stoker

● Since then, Faculty at the Courant Institute of Mathematical Sciences, New York University. He retired 1999


Richard Courant

at the Courant Institute of Mathematical Sciences (New York University)

Founded: 1935 Current Building: 1965


Kurt O. Friedrichs

Jürgen Moser

Courant Institute Directors after Richard Courant:

L. Nirenberg (left) and Peter Lax (right; Abel Prize 2005) S.R. Snirivasa Varadhan

(Abel Prize 2007)

http://cims.nyu.edu/webapps/content/about/history

http://www.cims.nyu.edu/gallery/

Allyn Jackson, 2002, “Interview with Louis Nirenberg”





James J. Stoker (PhD advisor) and Louis Nirenberg

Kurt O. Friedrichs (left) and Richard Courant (right)




L. Nirenberg's main awards and honors:

● the American Mathematical Society’s Bôcher Prize in 1959

● the Jeffrey-Williams Prize of the Canadian Mathematical Society in 1987

● the Steele Prize of the AMS in 1994 for Lifetime Achievement

● First recipient in mathematics of the Crafoord Prize, in 1982, established by the Royal Swedish Academy of Sciences in areas not covered by the Nobel Prizes. He shared the award with Vladimir Arnold

● Inaugural Chern Medal, in 2010, given by the International Mathematical Union and the Chern Medal Foundation

● Abel Prize 2015, with John F. Nash, Jr.

Receiving the Crafoord Prize, Stockholm 1982, with Mrs. Crafoord and the King of Sweden



Some important mathematical contributions of L. Nirenberg:

● Differential Geometry:

The Weyl problem; the Minkowski problem

● Complex Analysis:

Newlander-Nirenberg theorem; complex Monge-Ampère equations

● Real Analysis:

The BMO space; degree of VMO maps

● Theory of PDEs:

Regularity theory (inequalities and estimates); Monge-Ampère and fully nonlinear equations; free boundaries; Navier-Stokes equations; symmetry theorems (the moving planes and sliding methods); maximum principles; front propagation; etc, etc.

Partial Differential Equations. Types :

1. Elliptic : Laplace equation:

2. Parabolic :

● Heat or diffusion equation:

● Navier-Stokes (or 1 million $) equations

(incompressible viscous fluids)

3. Hyperbolic :

● Wave equation (acoustics, sound-waves)

● Schrödinger equation (quantum mechanics)

● Euler's equations (incompressible fluids)



Nirenberg's PhD Thesis (published in the above paper) solves an

Open Problem of H. Weyl from 1916:

Given a smooth metric g of positive curvature on the sphere S²,

is there an embedding X: S² R³ such that the metric induced on S² by this

embedding is g?


Soap films = Minimal surfaces




Dirichlet integral



Dirichlet integral


The Laplacian coming from physics:

heat, concentrations, gravitational and electrical potentials, viscous fluids, etc.

Fourier law for heat

Stationary solutionsThe heat equation

Pierre-Simon, marquis de Laplace (1749-1827)

Jean-Baptiste Joseph Fourier (1768-1830)

Pierre-Simon, marquis de Laplace (1749-1827)

The Laplacian coming from Finance and Probability:

what is your expected gain when,

starting always from the same given tile in your living room, you walk randomly and you get 30€ only when you hit a radiator on the first time that you hit your living room's walls (otherwise you get 0€)?

30€

30€

30€

0€

0€

0€

0€


How to solve the problem:

● make a squared lattice of very small step-size h

● Move from a point to either East, West, North, or South,

each one with probability 1/4

30€

30€

30€

0€

0€

0€

0€

N

S

EWC

C = starting point of the walk

u(C) = expected gain starting from C

(average)


h = step size of the lattice

The LAPLACIAN of u = 0 The LAPLACIAN of u = 0


● X. Cabré, Partial differential equations, geometry and stochastic control, in Catalan. Butl. Soc. Catalana Mat. 15 (2000), 7-27● X. Cabré, Elliptic PDEs in Probability and Geometry. Symmetry and regularity of solutions. Discrete Contin. Dyn. Syst. 20 (2008), 425-457


Rellich–Kondrachov theorem, an extension of

Arzelà-Ascoli theorem:

Poincaré inequality:

Existence theoremscome from estimates, whichgive compactness:








+ initial (and boundary) conditions


if n=2

if n=3

Sobolev-Gagliardo-Nirenberg inequalities (u=0 on boundary of B_r):



Back to Geometry:

Minimal surface equation for graphs:

Area functionalof films, graphs:


Back to Geometry:

K(x) = scalar curvatureu(x) = conformal factorThe Yamabe problem

K(x) = Gauss curvatureSurface = graph of u


solves an Open Problem of H. Weyl from 1916:

Given a smooth metric g of positive curvature on the sphere S², is there an embedding X: S² R³ such that the metric induced on S² by this embedding is g?

● Continuity method and IFT (implicit function theorem)

● Need estimates (regularity) for solutions of Monge-Ampère type equations in dimension 2. This gives that the linearized problem is an isopmorphism (IFT ok) but in HÖLDER or SOBOLEV spaces,

NOT from C² to C :⁰

with zero Dirichlet boundary conditionsis not an isomorphism if n > 1


solves an Open Problem of H. Weyl from 1916:

Given a smooth metric g of positive curvature on the sphere S², is there an embedding X: S² R³ such that the metric induced on S² by this embedding is g?

● Continuity method and IFT (implicit function theorem)

● Need estimates (regularity) for solutions of Monge-Ampère type equations in dimension 2. This gives that the linearized problem is an isopmorphism (IFT ok)

● Similarity with Perelman's proof of the Poincaré conjecture: Homotopy with a nonlinear geometric heat equation: the Ricci flow + estimates for analysis of singularities

● Estimates easier in dim 2 (complex variables, harmonic and analytic functions, quasiconformal mappings): work of C. B. Morrey


Towards estimates and regularity:

differentiating the equation (or making difference quotients: Nirenberg's method)

● Quasilinear equations:



L. Nirenberg's modesty and sense of humor:

De Giorgi-Nash-Moser Theorem: Hölder regularity of solutions of

with a_{ij} uniformly elliptic (positive definite matrices) but only bounded and measurable as a function of x in R^n.

● Nash, J. Parabolic equations. Proc. Nat. Acad. Sci. U.S.A. 43 (1957), 754-758.

● Nash, J. Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80 (1958), 931-954.

''A gold mine'', in Nirenberg's words.

Nash work retaken and presented in:

● Fabes, E. B.; Stroock, D. W. A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash. Arch. Rational Mech. Anal. 96 (1986), no. 4, 327-338.

Independently proved by:

● De Giorgi, Ennio. Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari. (Italian) Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 1957 25–43.

and later a new (third) proof by Jürgen Moser



Complex Analysis:

● A. Newlander and L. Nirenberg, Complex analytic coordinates in almost-complex manifolds, Ann. Of Math. 65 (1957), 391–404

solves the problem on integrability of almost complex structures. It was suggested to Nirenberg by A. Weil and S.S. Chern. When can one reduce a given system of n first order linear PDEs in R^2n to the Cauchy-Riemann equations in C^n, after a smooth change of variables.

Also an extension to a complex form of the classical Frobenius theorem on differential forms.

Foundation stone for:

● K. Kodaira, L. Nirenberg, and D. C. Spencer, On the existence of deformations of complex analytic structures, Ann. of Math. 68 (1958), 450–459

establishes the existence of deformations of complex structures on complex manifolds


Real Analysis:

The BMO space of

John-Nirenberg

Fritz John (Courant Institute)


Real Analysis:

The BMO space of

John-Nirenberg

Fritz John (Courant Institute)

The Hardy space H¹ :

● H¹ * = BMO

● VMO* = H¹

● Brezis-Nirenberg: degree theory of VMO maps



sequence of papers that establish a long time open

C^{2,\alpha} estimate up to the boundary for

solutions of the Monge-Ampère problem

● Independently proved ny N.V. Krylov (1983) ● Independently proved ny N.V. Krylov (1983)


sequence of papers that establish a long time open

C^{2,\alpha} estimate up to the boundary for

solutions of the Monge-Ampère problem

● Independently proved ny N.V. Krylov (1983) ● Independently proved ny N.V. Krylov (1983)

Nirenberg with Irene Gamba and Luis Caffarelli

Joel Spruck


Related to the study of bubbles in the Yamabe problem:

K(x) = scalar curvature for a new conformal metric

u(x) = conformal factor


Quoting L. Nirenberg:

''I made a living off the maximum principle''


Thus, u and v (solutions of the same nonlinear problem) such thatu \leq v and u(x_0)=v(x_0), then u \equiv v

If u and v are solutions of the same elliptic nonlinear equation,then u - v satisfies a linear elliptic equation with variable coefficients:

Two different soap films or soap bubbles cannot touch tangently


Proved with the moving planes method of A.D. Alexandrov, who created it to prove:

every embedded connected hypersurface in R^n with constant mean curvature must be a ball.


Henri Berestycki (Paris 6; now EHESS)

● The sliding method of Berestycki-Nirenberg. Leads to uniqueness, monotonicity of solutions, and Liouville type theorems

● The maximum principle on non smooth domains:


References:

● https://www.simonsfoundation.org/science_lives_video/louis-nirenberg/

(video of J. Shatah's interview to Louis Nirenberg)

● http://www.cims.nyu.edu/gallery/

● http://cims.nyu.edu/webapps/content/about/history

● Allyn Jackson, April 2002, “Interview with Louis Nirenberg,” Notices of the AMS, 49 (4), 441-449

● Simon Donaldson, March 2011, “On the Work of Louis Nirenberg,” Notices of the American Mathematical Society, 58 (3), 469-472

● Yan Yan Li, 2010, “Work of Louis Nirenberg,” Proceedings of the International Congress ofMathematicians, Hyderabad, India

● http://www.claymath.org/millenium-problems/navier%E2%80%93stokes-equation

(video by Luis Caffarelli's talk on Navier-Stokes)

● X. Cabré, Partial differential equations, geometry and stochastic control, in Catalan. Butl. Soc. Catalana Mat. 15 (2000), 7-27

● X. Cabré, Elliptic PDEs in Probability and Geometry. Symmetry and regularity of solutions. Discrete Contin. Dyn. Syst. 20 (2008), 425-457

https://www.simonsfoundation.org/science_lives_video/louis-nirenberg/



http://www.claymath.org/millenium-problems/navier%E2%80%93stokes-equation

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