ON THE REGULARIZATION OF THE CAUCHY PROBLEM FOR THE LAPLACE EQUATION IN THE STRIPE

Abstract

This paper investigates the problem of determining the temperature distribution along the upper boundary of a strip, given prescribed temperature data on its lower boundary. The analysis is carried out within the framework of the Laplace equation, leading to the consideration of a corresponding Cauchy-type problem. The uniqueness of the solution is rigorously established in an appropriate space of generalized functions. Furthermore, quantitative estimates are derived that characterize the relationship between solutions of the associated well-posed and ill-posed problems. On the basis of these estimates, approximation results are obtained, and relevant approximation theorems are proved, providing a theoretical foundation for stable reconstruction of the solution.