BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

On the Cherkaev-Gibiansky Method and its Applications: Bounds on Schur Complements of Dissipative Operators via Minimization Variational Principles Welters, Aaron


In this talk, I will discuss the Cherkaev-Gibiansky variational method for developing bounds on effective operators from the perspective of the abstract theory of composites and then apply the method to a large class of operators arising in the constitutive relations which include dissipative, coercive, and sectorial operators. The important points that will be made in this talk are: 1) the bounds are derived from variational minimization principles even for operators which are non-self-adjoint; 2) they apply to Schur complements of such operators as they are effective operators; 3) effective operators can be seen as a generalization of the Schur complement concept; 4) effective operators derived from Herglotz functions are themselves Herglotz functions and the variational bounds apply to such effective operators. To demonstrate the applicability of the method, we will discuss two important examples, the effective complex conductivity in the theory of composites and the Dirichlet-to-Neumann (DtN) map for the conductivity equation with a non-symmetric conductivity tensor on a Lipschitz bounded domain in two- or -three dimensions. This is joint work with Maxence Cassier (Institut Fresnel).

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