Abstract:
The minimal surface area of graphs of W^{1,1}-maps is a principal example of a degenerate elliptic variational problem as the existence and regularity of minimisers cannot be extracted from the direct method. To address such issues it is necessary to contemplate a relaxation of the problem to measure functionals and study the notion generalised minimisers which minimise the original functional.
In this talk we present the higher Sobolev regularity results for convex variational integrals of linear growth depending on vectorial differential operators A. This can be formulated as a version of Dirichlet problem posed on BV^{A}, the functions of bounded A-variation. Within our considerations is the family of operators A for which the nullspace is finite-dimensional and we shall discuss the importance of this very assumption in the given context. The key strategy to access regularity is to utilise a good viscosity approximation sequence which is obtained from the Ekeland Variational Principle on an appropriate perturbation space. We will outline the main techniques and constructions involved in the proof.
The exhibited work is a generalisation of the Sobolev regularity criteria proved for the cases of BV and BD.