Abstract
We consider the following data completion problem for the Laplace equation in the cylindrical domain: ( is a smooth bounded open set and a > 0), limited by the faces and . The Neumann and Dirichlet boundary conditions are given on Γ0 while no condition is given on Γa. The completion data problem consists in recovering a boundary condition on Γa. This problem has been known to be ill-posed since Hadamard [12]. The problem is set as an optimal control problem with a regularized cost function. To obtain directly an approximation of the missing data on Γa we use the method of factorization of elliptic boundary value problems. This method allows us to factorize a boundary value problem in the product of two parabolic problems. Here it is applied to the optimality system (i.e. jointly on the state and adjoint state equations).
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