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Part explanation of important recent work, and part introduction to some of the techniques of modern partial differential equations, this monograph is a self-contained exposition of the Neumann problem for the Cauchy-Riemann complex and certain of its applications. The authors prove the main existence and regularity theorems in detail, assuming only a knowledge of the basic theory of differentiable manifolds and operators on Hilbert space. They discuss applications to the theory of several complex variables, examine the associated complex on the boundary, and outline other techniques relevant to these problems. In an appendix they develop the functional analysis of differential operators in terms of Sobolev spaces, to the extent it is required for the monograph.

Functional analysis --- Neumann problem --- Differential operators --- Complex manifolds --- Complex manifolds. --- Differential operators. --- Neumann problem. --- Differential equations, Partial --- Équations aux dérivées partielles --- Analytic spaces --- Manifolds (Mathematics) --- Operators, Differential --- Differential equations --- Operator theory --- Boundary value problems --- A priori estimate. --- Almost complex manifold. --- Analytic function. --- Apply. --- Approximation. --- Bernhard Riemann. --- Boundary value problem. --- Calculation. --- Cauchy–Riemann equations. --- Cohomology. --- Compact space. --- Complex analysis. --- Complex manifold. --- Coordinate system. --- Corollary. --- Derivative. --- Differentiable manifold. --- Differential equation. --- Differential form. --- Differential operator. --- Dimension (vector space). --- Dirichlet boundary condition. --- Eigenvalues and eigenvectors. --- Elliptic operator. --- Equation. --- Estimation. --- Euclidean space. --- Existence theorem. --- Exterior (topology). --- Finite difference. --- Fourier analysis. --- Fourier transform. --- Frobenius theorem (differential topology). --- Functional analysis. --- Hilbert space. --- Hodge theory. --- Holomorphic function. --- Holomorphic vector bundle. --- Irreducible representation. --- Line segment. --- Linear programming. --- Local coordinates. --- Lp space. --- Manifold. --- Monograph. --- Multi-index notation. --- Nonlinear system. --- Operator (physics). --- Overdetermined system. --- Partial differential equation. --- Partition of unity. --- Potential theory. --- Power series. --- Pseudo-differential operator. --- Pseudoconvexity. --- Pseudogroup. --- Pullback. --- Regularity theorem. --- Remainder. --- Scientific notation. --- Several complex variables. --- Sheaf (mathematics). --- Smoothness. --- Sobolev space. --- Special case. --- Statistical significance. --- Sturm–Liouville theory. --- Submanifold. --- Tangent bundle. --- Theorem. --- Uniform norm. --- Vector field. --- Weight function. --- Operators in hilbert space --- Équations aux dérivées partielles