Abstract

The authors consider the problem of reconstructing the temperature of a body from the interior measurements. In fact, they study a semilinear elliptic PDE \(\Delta u=f(x,u)\) defined in the upper-half of the plane with the Dirichlet condition on the boundary and they seek for a solution that vanishes at infinity or the same problem defined on a strip with mixed Dirichlet and Neumann conditions. Using Green functions, the authors transform the original problem to integral equations, show the existence of solutions and give an effective way to approximate the actual value of partial derivatives of solutions. Then, by the use of the method of the so-called truncated high frequencies of Fourier series, the problem is suitably regularized, the existence of regularized solutions is established by means of the contraction principle, and the error bounds are estimated.