CAUCHY PROBLEM FOR THE BIHARMONIC EQUATION

Abstract

In this paper, we study the problem of continuing the solution of the Cauchy problem for a biharmonic equation in a domain by its known values ​​on the smooth part of the boundary. The problem under consideration belongs to the problems of mathematical physics, in which there is no continuous dependence of solutions on the initial data. It is assumed that a solution to the problem exists and is continuously differentiable in a closed domain with exactly given Cauchy data. For this case, using the Carleman function, an explicit regularization formula is proposed. It is assumed that the solution is bounded on a part of the boundary. The method for obtaining results is based on the construction of constructing a fundamental solution of the Laplace equation in an explicit form, depending on a positive parameter that tends to zero as the parameter tends to infinity on the part of the boundary of the region in which conditional Cauchies are not given.