Global Regularity of the ∂-Neumann Problem: A Survey of the L²-Sobolev Theory

Summary

The fundamental boundary value problem in the function theory of several complex variables is the ∂-Neumann problem. The L² existence theory on bounded pseudoconvex domains and the C^∞ regularity of solutions up to the boundary on smooth, bounded, strongly pseudoconvex domains were proved in the 1960s. On the other hand, it was discovered quite recently that global regularity up to the boundary fails in some smooth, bounded, weakly pseudoconvex domains. We survey the global regularity theory of the ∂-Neumann problem in the setting of L² Sobolev spaces on bounded pseudoconvex domains, beginning with the classical results and continuing up to the frontiers of current research. We also briefly discuss the related global regularity theory of the Bergman projection.

1. Introduction

The ∂-Neumann problem is a natural example of a boundary-value problem with an elliptic operator but with non-coercive boundary conditions. It is also a prototype (in the case of finite-type domains) of a subelliptic boundary-value problem, in much the same way that the Dirichlet problem is the archetypal elliptic boundary-value problem. In this survey, we discuss global regularity of the ∂-Neumann problem in the L²-Sobolev spaces W^s(Ω) for all non-negative s and also in the space. For estimates in other function spaces, such as Holder spaces and L^p-Sobolev spaces, see [Beals et al. 1987; Berndtsson 1994; Chang et al. 1992; Cho 1995; Christ 1991; Fefferman 1995; Fefferman and Kohn 1988; Fefferman et al. 1990; Greiner and Stein 1977; Kerzman 1971; Krantz 1979; Lieb 1993; McNeal 1991; McNeal and Stein 1994; Nagel et al. 1989; Sträube 1995]; for questions of real analytic regularity, see, for example, [Chen 1988; Christ 1996b; Derridj and Tartakoff 1976; Komatsu 1976; Tartakoff 1978; 1980; Tolli 1996; Treves 1978] and [Christ 1999, Section 10] in this volume.