Efficient Construction Technique for Resolving Dirichlet Boundary-Value Problems in Multiply Connected Domains for Real Elliptic Equations

This paper introduces a novel analytic approach for solving Dirichlet boundary-value problems for regular solutions of second-order elliptic differential equations. The methodology uses complex variables \(z=x+iy\) and \(\zeta =x-iy\) and integral representation techniques for regular solutions within the multiply connected domain \(T\) . The regular solution is represented as \(\text{Re} U(z, \zeta )\) satisfying the symmetry condition \(U(\overline{\zeta }, \overline{z } ) = \stackrel{-}{(U(z, \zeta ))}\) . This approach extends its applicability beyond domains bounded by Lyapunov and Vekua and provides efficient resolutions for Dirichlet boundary-value problems in multiply connected domains for real elliptic equations.