# Groundbreaking Contributions to the Theory of Nonlinear PDEs

In Oslo on May 19, His Majesty King Harald of Norway will present the 2015 Abel Prize to John F. Nash, Jr., of Princeton University and Louis Nirenberg of the Courant Institute of Mathematical Sciences, New York University. In announcing the 2015 awards, the Norwegian Academy of Science and Letters cited the laureates jointly “for striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis.”

An extended citation, part of a wealth of material available on the Abel Prize website, opens with a succinct overview of the laureates’ contributions to the theory of nonlinear PDEs:

*“Nash and Nirenberg have played a leading role in the development of this theory, by the solution of fundamental problems and the introduction of deep ideas. Their breakthroughs have developed into versatile and robust techniques, which have become essential tools for the study of nonlinear partial differential equations. Their impact can be felt in all branches of the theory, from fundamental existence results to the qualitative study of solutions, both in smooth and non-smooth settings. Their results are also of interest for the numerical analysis of partial differential equations.”*

Highlights of their contributions include:

- John Nash in 1994. Photo courtesy of Princeton University, Office of Communications.
**Fundamental results in elliptic regularity.**Nash proved the first Hölder estimates for solutions of linear elliptic equations in general dimensions with no regularity assumptions on the coefficients (a result also proved by De Giorgi around the same time by an entirely different method). Nirenberg’s many results on elliptic regularity include his systematic study (with Agmon and Douglis) of elliptic boundary value problems with \(L^p\) data. **Transformational results in geometric analysis.**Nash’s smooth \((C^\infty)\) isometric embedding theorem established the equivalence of Riemann’s intrinsic point of view with the older extrinsic approach; his non-smooth \((C^1)\) isometric embedding theorem demonstrated the possiblilty of embeddings seemingly forbidden by such geometric invariants as Gauss curvature. Nirenberg’s contributions to geometric analysis include his earliest work, in which he solved problems posed by Minkowski and Weyl concerning the realizability of metrics on \(S^2\) with positive curvature by convex surfaces in \(R^3\); of comparable importance is the Newlander–Nirenberg theorem, a fundamental result in complex differential geometry.**New methods and crucial tools.**Among them in Nash’s case are the Nash–DeGiorgi–Moser regularity theory and the Nash–Moser inverse function theorem; for Nirenberg they include the development (with Joseph Kohn) of the theory of pseudodifferential operators. Also noteworthy are crucial inequalities—including the the John–Nirenberg inequality (which laid the foundation for a new chapter in harmonic analysis involving duality between the space of functions of bounded mean oscillation and the Hardy space \(H^1\)), the Gagliardo–Nirenberg interpolation inequalities (an extremely useful generalization of the Sobolev inequalities), and Nash’s inequality (first proved by Stein, important among other reasons for the study of probabilistic semigroups).**And much, much more.**In Nirenberg’s case, another very influential theme was development (with Gidas and Ni) of the “method of moving planes,” a powerful technique for demonstrating that solutions of certain elliptic problems must share the symmetries of the problem. Another of Nash’s striking contributions was the realization of manifolds as algebraic varieties.

“Towering figures” individually in the analysis of PDEs, “Nash and Nirenberg influenced each other through their contributions and interactions,” the Abel Prize citation concludes. “The consequences of their fruitful dialogue, which they initiated in the 1950s at the Courant Institute of Mathematical Sciences, are felt more strongly today than ever before.”

*Nirenberg was the first (2010) recipient of the Chern Medal, which is awarded every four years at the International Congress of Mathematicians. In 1994, Nash was one of three recipients of the Nobel Prize in Economic Sciences for work on the theory of non-cooperative games. *

*For a deeper and more informal perspective on Nash’s and Nirenberg’s lives and work, the Simons Foundation’s Science Lives interviews are a good resource. Nirenberg was interviewed in 2010, and again in 2011, by Jalal Shatah; Charles Fefferman and the late Harold Kuhn interviewed Nash in 2011. *