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|Title:||Factorization by invariant embedding of elliptic problems: circular and star-shaped domains|
|Author:||Soares, Maria do Céu Cerqueira|
|Publisher:||FCT - UNL|
|Abstract:||This thesis concerns the factorization of elliptic operators, namely the decomposition of a second order boundary value problem, de¯ned in an open bounded regular domain, in an uncoupled system of two ¯rst order initial value problems. The method presented here is inspired on the theory of Optimal Control. It is a return, in a new spatial approach, to the technique of the invariant temporal embedding, de¯ned originally in the context of Dynamic Programming, used in Control Theory for the computation of the optimal feedback. This technique consists in embedding the initial problem in a family of similar problems depending on a parameter, which are solved recursively. In our case, each problem is de¯ned over a sub-domain limited by a mobile boundary depending on the parameter. We introduce an operator relating the trace of the function de¯ned for each problem, and the trace of its normal derivative over the mobile boundary. Without loss of generality, we particularize the study to a Poisson's equation with, for example, a Dirichlet's boundary condition. We ¯rst consider a circular domain and we present for it two approaches: ¯rst, we apply an invariant embedding that starts on the boundary of the circle and go towards its center, followed by an invariant embedding in the opposite direction. Next, we generalize the method, applying it to the case of an arbitrary star shaped domain. In all cases, the family of curves which limits the subdomains de¯ned by the invariant embedding are homothetic to one another and homothetic to a point. This fact induces the appearing of a singularity.|
|Description:||Dissertação apresentada para obtenção do grau de Doutor em Matemática na especialidade de Equações Diferenciais, pela Universidade Nova de Lisboa,Faculdade de Ciências e Tecnologia|
|Appears in Collections:||FCT: DM - Teses de Doutoramento|
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