point theory dealt with local minimizers. 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. Their work was later extended by Jesús García Azorero, Juan Manfredi, and Ireneo Peral. 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. With Berestycki and Italo Capuzzo-Dolcetta, Nirenberg studied superlinear equations of Yamabe type, giving various existence and non-existence results. Nonlinear functional analysis 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. 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. 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. 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. 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. His work was later simplified by Takaaki Nishida and used in an analysis of the Boltzmann equation. Geometric problems Making use of his work on fully nonlinear elliptic equations, Nirenberg's Ph.D. thesis provided a resolution of the Weyl problem and Minkowski problem in the field 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 for which the Gauss map prescribes the Gaussian curvature. The