Abstract
The techniques and results contained in this monograph arose as, and from, the solution of a long-standing problem on the interface between complex analysis and partial differential equations, namely the (local) analytic hypoellipticity of the “\(\overline{\partial }\)-Neumann problem” at the boundary of a strictly pseudoconvex domain in \({\mathbb{C}}^{n}.\) This problem was introduced in its modern form by J.J. Kohn in [K1], where he proved local C ∞ hypoellipticity by way of what amounted, in modern language, to a “subelliptic” estimate for test functions satisfying the so-called \(\overline{\partial }\)-Neumann boundary conditions (\(v \in \mathcal{D}({\overline{\partial }}^{{_\ast}})\)). This led to the canonical solution of \(\overline{\partial }\) on such domains and hence to the solution of the Cousin problem concerning domains of holomorphy. Another approach to the solution of the Cousin problem had been used by Hörmander [Hö4] but is of quite a different character, not involving boundary regularity.
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Tartakoff, D.S. (2011). Brief Introduction. In: Nonelliptic Partial Differential Equations. Developments in Mathematics, vol 22. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9813-2_2
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DOI: https://doi.org/10.1007/978-1-4419-9813-2_2
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