Compl. anal. oper. theory 4 (2010), 39–53 c  2008 Birkh¨auser Verlag Basel/Switzerland 1661-8254/010039-15, published online October 13, 2008 DOI 10.1007/s11785-008-0086-6 Complex Analysis and Operator Theory Solutions of Robin Problems for Overdetermined Inhomogeneous Cauchy–Riemann Systems on the Unit Polydisc Alip Mohammed and M. W. Wong Abstract. The inhomogeneous Robin condition with general coefficient for the overdetermined inhomogeneous Cauchy–Riemann system of equations on the polydisc is studied using Fourier analysis. It is shown that this problem for the case of nonholomorphic general coefficient, is actually a problem with essential singularity in the domain, but still well-posed under certain compatibility conditions. Under proper assumptions, the unique solution is given explicitly. Mathematics Subject Classification (2000). 32A10, 35C10, 35N10. Keywords. Robin problem, overdetermined inhomogeneous Cauchy–Riemann systems, essential singularities. 1. Introduction Let D n be the unit polydisc given by D n =  z =(z 1 ,z 2 ,...,z n ) ∈ C n : |z k | < 1, 1 ≤ k ≤ n  and let ∂ 0 D n be its essential boundary given by ∂ 0 D n =  z =(z 1 ,z 2 ,...,z n ) ∈ C n : |z k | =1, 1 ≤ k ≤ n  . We are interested in finding a function v in C 1 (D n ) such that ⎧ ⎨ ⎩ ∂v ∂ z k = f k (z) , z ∈ D n , ∂v ∂ν ζ + α(ζ )v = γ 0 (ζ ) , ζ ∈ ∂ 0 D n , (1.1) This research is supported by the Natural Sciences and Engineering Research Council of Canada. Alip Mohammed is on leave from Department of Mathematics, Xinjiang University, Shenglilu 14, 830046 ¨ Ur¨ umqi, PR China.