ON THE REGULARIZATION OF THE CAUCHY PROBLEM FOR MATRIX FACTORIZATIONS OF THE HELMHOLTZ EQUATION IN Rm

Abstract

In this paper, we delve into the intricate challenge of extending solutions to the ill-posed Cauchy problem linked to matrix factorizations of the Helmholtz equation, set within both bounded and unbounded multidimensional domains. We presuppose the existence of a solution that is continuously differentiable throughout the entire closed domain, anchored by the
specified Cauchy data. Given these conditions, we derive explicit formulas for the extension of this solution alongside a robust regularization method. Our proposed solutions encompass continuous approximations that faithfully conform to a predetermined error measure within the uniform metric, effectively replacing the original Cauchy data. Furthermore, this study offers an estimation of the stability of the solution to the Cauchy problem, framed within a classical context. Through this exploration, we not only aim to advance the mathematical understanding of the Helmholtz equation but also to illuminate pathways for practical applications wherein such extended solutions can be effectively utilized.