Factorization of elliptic boundary value problems by invariant embedding and application to overdetermined problems

Dissertação para obtenção do Grau de Doutor em Matemática

The purpose of this thesis is the factorization of elliptic boundary value problems defined in cylindrical domains, in a system of decoupled first order initial value problems. We begin with the Poisson equation with mixed boundary conditions, and use the method of invariant embedding: we embed our initial problem in a family of similar problems, defined in sub-domains of the initial domain, with a moving boundary, and an additional condition in the moving boundary. This factorization is inspired by the technique of invariant temporal embedding used in Control Theory when computing the optimal feedback, for, in fact, as we show, our initial problem may be defined as an optimal control problem. The factorization thus obtained may be regarded as a generalized block Gauss LU factorization. From this procedure emerges an operator that can be either the Dirichlet-to-Neumann or the Neumann-to-Dirichlet operator, depending on which boundary data is given on the moving boundary. In any case this operator verifies a Riccati equation that is studied directly by using an Yosida regularization. Then we extend the former results to more general strongly elliptic operators. We also obtain a QR type factorization of the initial problem, where Q is an orthogonal operator and R is an upper triangular operator. This is related to a least mean squares formulation of the boundary value problem. In addition, we obtain the factorization of overdetermined boundary value problems, when we consider an additional Neumann boundary condition: if this data is not compatible with the initial data, then the problem has no solution. In order to solve it, we introduce a perturbation in the original problem and minimize the norm of this perturbation, under the hypothesis of existence of solution. We deduce the normal equations for the overdetermined problem and, as before, we apply the method of invariant embedding to factorize the normal equations in a system of decoupled first order initial value problems.