At the beginning of the twentieth century, a major question of several complex variables was the classification of domains of holomorphy. It was known that the domains of holomorphy were pseudoconvex; the converse however, became known as the Levi problem. In the middle of the twentieth century Oka first solved the problem and therefore showed that the domains of holomorphy were precisely the pseudoconvex domains. In the decades following this result, many different methods of studying the Levi problem were developed. In this thesis, we focus on the method of partial differential equations for solving the nonhomogeneous Cauchy-Riemann equation developed by Hörmander.
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