On the error estimate of gradient inclusions

The numerical analysis of gradient inclusions in a compact subset of $2\times 2$ diagonal matrices is studied. Assuming that the boundary conditions are reached after a finite number of laminations and using piecewise linear finite elements, we give a general error estimate in terms of the number of laminations and the mesh size. This is achieved by reduction results from compact to finite case.Comment: 21 pages, 4 figure

Relaxation and 3d-2d passage with determinant type constraints: an outline

We outline our work (see [1,2,3,4]) on relaxation and 3d-2d passage with determinant type constraints. Some open questions are addressed. This outline-paper comes as a companion to [5]

On the relaxation of variational integrals in metric Sobolev spaces

We give an extension of the theory of relaxation of variational integrals in classical Sobolev spaces to the setting of metric Sobolev spaces. More precisely, we establish a general framework to deal with the problem of finding an integral representation for relaxed variational functionals of variational integrals of the calculus of variations in the setting of metric measure spaces. We prove integral representation theorems, both in the convex and non-convex case, which extend and complete previous results in the setting of euclidean measure spaces to the setting of metric measure spaces. We also show that these integral representation theorems can be applied in the setting of Cheeger-Keith's differentiable structure.Comment: 26 page