A theory of Lower Semicontinuity for Integral Functionals with Linear Growth and u-dependence, Giles Shaw (Reading & Cambridge)

By | October 3, 2016

Variational problems with linear growth arise naturally in the Calcu- lus of Variations from the study of singular perturbation problems associated to a large number of physical and mathematical applications. These problems must be posed over the class of functions of bounded variation, and their analysis is signifi- cantly more involved than that which is required for variational problems posed over Sobolev spaces. I will discuss my progress with Filip Rindler towards developing a general, robust, theory to study the lower semicontinuity properties of this class of problems. Our approach involves associating a class of measures to the graphs of BV functions before constructing a family of Young measures to characterise their limiting behaviour. On a technical level, we make use of fine properties of BV functions as well as methods from geometric measure theory.