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Compensated Compactness and L1-estimates

Dr. Bogdan Raita

Dienstag, 2021-01-19 14:00Uhr

In the first part of the talk we will review some recent results on the Murat--Tartar framework of Compensated Compactness Theory, by which we mean weak (lower semi-)continuity of nonlinear functionals interacting with weakly convergent sequences of PDE constrained vector fields. We present improvements of the original work of Murat and Tartar, as well as more recent work of Fonseca--Müller. We also present answers to questions of Coifman--Lions--Meyer--Semmes and De Philippis. The second part of the talk will concern properties of solutions of linear systems $ L u = \mu $, where $ \mu $ is a Radon measure, a borderline case not covered by Calderón--Zygmund Theory. We build on the fundamental work of Bourgain--Brezis-- Mironescu and Van Schaftingen towards surprising strong interior Sobolev estimates for solutions. We also discuss the start of a theory towards estimates up to the boundary. The final part of the talk will cover fine properties of solutions and possible intersections with Geometric Measure Theory.