An Integral Representation Formula for First-Order Elliptic Systems with Constant Coefficients in Layer-Type Domains

This paper investigates the validity of an integral representation formula for a vector function satisfying a first-order elliptic system with constant coefficients in an unbounded domain of layer type. The study builds upon earlier results concerning bounded domains, extending the applicability of such formulas to broader unbounded settings. Drawing from foundational works by Tarkhanov, Yarmukhamedov, and Bitsadze, the authors present a rigorous framework to validate the integral formula under growth conditions imposed on the solution and the domain boundary. The proof is structured through the introduction of specially constructed entire functions and asymptotic estimates, which guarantee the convergence of the integral representation. Two theorems are proved: the first establishes the formula’s validity for vector functions with controlled growth inside the domain; the second generalizes the result to more inclusive function classes with boundary growth conditions. The results contribute to the theory of elliptic systems by providing tools for solving boundary value problems in more complex, unbounded geometries.