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CNA Seminar/Colloquium/Joint Pitt-CNA Colloquium

Giles Shaw
University of Cambridge
Title: Continuity and Lower Semicontinuity for u-Dependent Functionals over BV

Abstract: A famous family of problems in the Calculus of Variations is to find necessary and sufficient conditions for the lower semicontinuity of integral functionals associated to integrands f=f(x,u,Du) which satisfy a p-growth condition in the final variable. The most challenging member of this family is the case p=1, where one is forced to seek a minimizer in the space of functions of bounded variation. Challenges here include defining the minimization problem correctly over BV in the first place, the lack of equiintegrability which minimizing sequences can now display, and the fact that one must deal with the pointwise behaviour of a function on a set where its derivative concentrates.

The most general result in this area is due to Fonseca and Müller who showed that, as for p>1, quasiconvexity is the right condition for lower semicontinuity. Their result requires several continuity and boundedness hypotheses which are not needed for the case p>1, and it is not clear which of these are truly an unavoidable feature of the p=1 regime and which might be omitted. I will discuss my progress with Filip Rindler towards understanding which of these hypotheses can be removed and in creating a new, cleaner, proof of Fonseca and Muller's result. Our approach involves associating a class of measures to the graphs of BV functions before constructing a family of Young measures to characterise the limiting behaviour of these objects. On a technical level, we make use of fine properties of BV functions as well as tangent measures from geometric measure theory."

Date: Tuesday, May 12, 2015
Time: 1:30 pm
Location: Wean Hall 7218
Submitted by:  David Kinderlehrer