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Analysis of the \(\overline\partial\)-Neumann problem along a straight edge. (English) Zbl 1057.32015

In a previous article [Indiana Univ. Math. J. 52, No. 3, 629–666 (2003; Zbl 1056.32024)], the author studied singularities of solutions of the \(\overline{\partial}\)-Neumann problem (for data in the Schwartz space) on the product of two half-planes. Now the author gives a generalization to the case of domains of the form \(\{(z_1,z_2)\in {\mathbb C}^2: \operatorname{Im} z_1 > \alpha \operatorname{Im} z_2,\) \(\operatorname{Im} z_2 >0\}\), where \(\alpha\geq0\). The main results are that there is a solution that belongs to the space \(L^p\) when \(p>2\), the solution is smooth away from the edge, and the singularities that arise at the edge have logarithmic and arctangent forms.

MSC:

32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
35B65 Smoothness and regularity of solutions to PDEs
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs

Citations:

Zbl 1056.32024
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