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SIMBa: Partial boundary value problems on finite networks

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El proper dilluns, 26 de novembre, se celebrarà una nova sessió del Seminari Informal de Matemàtiques de Barcelona (SIMBa).

Speaker: Cristina Araúz Lombardía.
Universitat: Departament de Matemàtica Aplicada III, Universitat Politècnica de Catalunya

Data: dilluns 26 de novembre de 2012
Hora: 12:15, cafè i galetes; 12:30, inici
Lloc: Aula IMUB (al terrat), Facultat de Matemàtiques de la Universitat de Barcelona.

Títol: Partial boundary value problems on finite networks
Resum: Inverse boundary-value problems were born to answer the question of whether it is possible to determine the conductivity of a body by means of boundary measurements. These problems are exponentially ill-posed since its solutions are highly sensitive to changes in the boundary data. We are mainly interested on the discrete version of the problem, that is, the inverse boundary-value problems on finite weighted networks. The aim here is to study partial inverse boundary-value problems, which are characterized by the existance of a part of the boundary where no data is known.
Given a weighted network with conductances on the edges \Gamma=(V,c), we fix a proper and connected subset F\subset V and will consider a certain kind of boundary value problems in which the values of the functions and of their normal derivatives are known at the same part of the boundary of F and there exists another part of the boundary where no data is known. We determine when there is existance and/or uniqueness of solution on \bar F. For, it is mandatory to consider the Dirichlet-to-Neumann map of the network, its kernel and a local inverse of the matrix given by this kernel. We also observe that the kernel of the Dirichlet-to-Neumann map is a Schur Complement of the Schrödinger operator of the network.
Joint work with Ángeles Carmona and Andrés M. Encinas.

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