Louis Nirenberg

Nirenberg's Ph.D. thesis provided a resolution of the Weyl problem and Minkowski problem of differential geometry. 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 of prescribed Gaussian curvature. The now-standard approach to these problems is through the theory of the Monge-Ampère equation, which is a fully nonlinear elliptic partial differential equation. Nirenberg made novel contributions to the theory of such equations in the setting of two-dimensional domains, building on the earlier 1938 work of Charles Morrey. Nirenberg's work on the Minkowski problem was significantly extended by Aleksei Pogorelov, Shiu-Yuen Cheng, and Shing-Tung Yau, among other authors. In a separate contribution to differential geometry, Nirenberg and Philip Hartman characterized the cylinders in Euclidean space as the only complete hypersurfaces which are intrinsically flat.

In the same year as his resolution of the Weyl and Minkowski problems, Nirenberg made a major contribution to the understanding of the maximum principle, proving the strong maximum principle for second-order parabolic partial differential equations. This is now regarded as one of the most fundamental results in this setting.[11]

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. In collaboration with Douglis and Shmuel Agmon, Nirenberg extended these estimates up to the boundary. With Morrey, Nirenberg proved that solutions of elliptic systems with analytic coefficients are themselves analytic, extending to the boundary earlier known work. 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.[12]

In 1957, 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 a precise condition under which an almost complex structure arises from a holomorphic coordinate atlas. 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.

In his 1959 survey on elliptic differential equations, Nirenberg proved (independently of Emilio Gagliardo) what is now known as the Gagliardo-Nirenberg interpolation inequalities for the Sobolev spaces. A later work by Nirenberg, in 1966, clarified the possible exponents which can appear in these inequalities. More recent work by other authors has extended the Gagliardo-Nirenberg inequalities to the fractional Sobolev spaces.

Immediately following Fritz John's introduction of the BMO function space in the theory of elasticity, John and Nirenberg gave a further study of the space, with a particular functional inequality, now known as the John-Nirenberg inequality, which has become basic in the field of harmonic analysis. It characterizes how quickly a BMO function deviates from its average; the proof is a classic application of the Calderon-Zygmund decomposition.

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. 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.

Nirenberg and J.J. 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.

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

${\displaystyle {\frac {du}{dt}}+Au=0}$

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

In the 1960s, A.D. Aleksandrov introduced an elegant "sliding plane" reflection method, which he used to apply the maximum principle in proving that the only closed hypersurface of Euclidean space which has constant mean curvature is the round sphere. In collaboration with Basilis Gidas and Wei-Ming Ni, Nirenberg gave an extensive study of how this method applies to prove symmetry of solutions of certain symmetric second-order elliptic partial differential equations. A sample result is that if u is a positive function on a ball with zero boundary data and with Δu + f(u) = 0 on the interior of the ball, then u is rotationally symmetric. In a later 1981 paper, they extended this work to symmetric second-order elliptic partial differential equations on all of ℝⁿ. These two papers are among Nirenberg's most widely cited, due to the flexibility of their techniques and the corresponding generality of their results. Due to Gidas, Ni, and Nirenberg's results, in many cases of geometric or physical interest, it is sufficient to study ordinary differential equations rather than partial differential equations. The resulting problems were taken up in a number of influential works by Ni, Henri Berestycki, Pierre-Louis Lions, and others.

Nirenberg and Charles Loewner studied the means of naturally assigning a complete Riemannian metric to bounded open subsets of Euclidean space, modeled on the classical assignment of hyperbolic space to the unit ball, via the unit ball model. They showed that if Ω is a bounded open subset of ℝ² with smooth and strictly convex boundary, then the Monge-Ampère equation

${\displaystyle \operatorname {det} D^{2}u={\frac {1}{|u|^{4}}}}$

has a unique smooth negative solution which extends continuously to zero on the boundary ∂Ω. The geometric significance of this result is that 1/−uD²u then defines a complete Riemannian metic on Ω. In the special case that Ω is a ball, this recovers the hyperbolic metric. Loewner and Nirenberg also studied the method of conformal deformation, via the Yamabe equation

${\displaystyle \Delta u=cu^{(n+2)/(n-2)}}$

for a constant c. They showed that for certain Ω, this Yamabe equation has a unique solution which diverges to infinity at the boundary. The geometric significance of such a solution is that u^{2/(n − 2)}g_Euc is then a complete Riemannian metric on Ω which has constant scalar curvature.

In other work, Haïm Brezis, Guido Stampacchia, and Nirenberg gave an extension of Ky Fan's topological minimax principle to noncompact settings. 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.

Luis Caffarelli, Robert Kohn, and Nirenberg studied the three-dimensional incompressible Navier-Stokes equations, showing that the set of spacetime points at which weak solutions fail to be differentiable must, roughly speaking, fill less space than a curve. This is known as a "partial regularity" result. In his description of the conjectural regularity of the Navier-Stokes equations as a Millennium prize problem, Charles Fefferman refers to Caffarelli-Kohn-Nirenberg's result as the "best partial regularity theorem known so far" on the problem. As a by-product of their work on the Navier-Stokes equations, Caffarelli, Kohn, and Nirenberg (in a separate paper) extended Nirenberg's earlier work on the Gagliardo-Nirenberg interpolation inequality to certain weighted norms.

In 1977, Shiu-Yuen Cheng and Shing-Tung Yau had resolved the interior regularity for the Monge-Ampère equation, showing in particular that if the right-hand side is smooth, then the solution must be smooth as well. In 1984, Caffarelli, Joel Spruck, and Nirenberg used different methods to extend Cheng and Yau's results to the case of boundary regularity. They were able to extend their study to a more general class of fully nonlinear elliptic partial differential equations, in which solutions are determined by algebraic relations on the eigenvalues of the matrix of second derivatives. With J.J. Kohn, they also found analogous results in the setting of the complex Monge-Ampère equation.

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.

The moving plane method of Aleksandrov, as extended in 1979 by Gidas, Ni, and Nirenberg, is studied further in joint works by Berestycki, Caffarelli, and Nirenberg. The primary theme is to understand when a solution of Δu+f(u)=0, with Dirichlet data on a cylinder, necessarily inherits a cylindrical symmetry.

In 1991, Brezis and Nirenberg applied the Ekeland variational principle to extend the mountain pass lemma. In 1993, they made a fundamental contributions to critical point theory in showing (with some contextual assumptions) that a local minimizer of

${\displaystyle \int _{\Omega }{\Big (}|\nabla u|^{2}-\int _{0}^{u(x)}f(x,t)\,dt{\Big )}\,dx}$

in the C¹ topology is also a local minimizer in the W^1,2 topology. In 1995, they used density theorems to extend the notion of topological degree from continuous mappings to the class of VMO mappings.

With Berestycki and Italo Capuzzo-Dolcetta, Nirenberg studied superlinear equations of Yamabe type, giving various existence and non-existence results. These can be viewed as developments of Brezis and Nirenberg's fundamental paper from 1983.

In an important result with Berestycki and Srinivasa Varadhan, Nirenberg extended the classically-known results on the first eigenvalue of second-order elliptic operators to settings where the boundary of the domain is not differentiable.

In 1992, Berestycki and Nirenberg gave a complete study of the existence of traveling-wave solutions of reaction-diffusion equations in which the spatial domain is cylindrical, i.e. of the form ℝ×Ω'.

With Yanyan Li, and motivated by composite materials in elasticity theory, Nirenberg studied 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.