Differential operators determining solutions of Elliptic equations

Abstract

We construct differential operatorsLg(z), Kg(z), Nf¯(z), Mf¯z) which map arbitrary functions holomorphic in a simply connected domainD of the planez=x+iy into regular solutions of the equation $$W_{z\bar z} + A(z,\bar z)W_{\bar z} + B(z,\bar z)W = 0$$ and present examples of the application of these differential operators to the solution of fundamental boundary-value problems in mathematical physics.