On the Cauchy problems for the Poisson equations with operator coefficients

We consider the Cauchy problems for operator analogues of the Poisson equations with data given in one case all around the boundary and the other on the part of the boundary. The problem is recast as an inverse boundary value problem for a system of coupled Beltrami-like equation associated with A -analytic and A -anti-analytic maps, which we solve by an extension of Bukhgeim’s A -analytic theory.