The classical and the modified Neumann problems for the inhomogeneous pluriholomorphic system in polydiscs (Q1578920)

The author studies the inhomogeneous system of \({n(n+ 1)\over 2}\) independent equations \[ {\partial^2u\over \partial\overline z_k\partial\overline z_l}= f_{kl}(z), 1\leq k,\;l\leq n, \] in the polydisc. It is clear that the right-hand side has to satisfy compatibility conditions. Some boundary value problems are considered: \(\bullet\) the Neumann problem for the above system, this means \({\partial u\over\partial\nu}= \gamma_0\) on \(\partial_0\mathbb{D}^n\), where \(\partial_0\mathbb{D}^n\) denotes the boundary of the polydisc and \({\partial\over\partial\nu}\) denotes the outward normal to \(\partial_0\mathbb{D}^n\); \(\bullet\) the Dirichlet-Neumann problem for the above system, this means \({\partial u\over\partial\nu}= \gamma_0\), \(\text{Re }u= \gamma\) on \(\partial_0\mathbb{D}^n\). For both problems the author derives \(\bullet\) necessary and sufficient conditions for the solvability; \(\bullet\) the structure of the solution for corresponding homogeneous problems; \(\bullet\) explicit representations of the solutions. To this known representation formula are used and the question is studied if a given continuous function on \(\partial_0\mathbb{D}^n\) describes the boundary behaviour of a holomorphic function in \(\mathbb{D}^n\).