Nirenberg's contributions to complex analysis (Q541554)

This article gives an overview over the fundamental contributions of Louis Nirenberg to complex analysis and their impact on the theory of partial differential equations. It explains the main results of Nirenberg and sketches the developments that they engendered. The author of the present paper is one of the real experts in the field and a main collaborator of Nirenberg. The material is put together in a nice order and gives a good overview of some main developments in complex analysis which are related to partial differential equations. Problems in complex analysis have often been studied with the aid of the theory of partial differential equations. Conversely, important advances in the theory of partial differential equations have come from the study of problems in complex analysis. In the last half century the interactions between these fields have been very fruitful and Louis Nirenberg was a leading participant in these developments. This article describes some of Nirenberg's contributions to complex analysis and some of his related work in partial differential equations. Topics covered are the Newlander-Nirenberg theorem and the Nirenberg-Frobenius theorem on integrability, applications of these to deformations of complex structures, the \(\overline{\partial}\)-Neumann problem (and subelliptic estimates), the theory of pseudodifferential operators, degenerate elliptic-parabolic equations, holomorphic support functions, local embeddings of pseudoconvex CR-manifolds, biholomorphic invariants, the complex Monge-Ampère equation.