Lower semicontinuity in the space BD of functions of bounded deformation

Lower semicontinuity in the space BD of functions of bounded deformation

Mon, 07/03/2011
17:00 		Filip Rindler (University of Oxford) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Lower semicontinuity in the space BD of functions of bounded deformation
The space BD of functions of bounded deformation consists of all L1-functions whose distributional symmetrized derivative (defined by duality with the symmetrized gradient ($ \nabla u + \nabla u^T)/2 $) is representable as a finite Radon measure. Such functions play an important role in a variety of variational models involving (linear) elasto-plasticity. In this talk, I will present the first general lower semicontinuity theorem for symmetric-quasiconvex integral functionals with linear growth on the whole space BD. In particular we allow for non-vanishing Cantor-parts in the symmetrized derivative, which correspond to fractal phenomena. The proof is accomplished via Jensen-type inequalities for generalized Young measures and a construction of good blow-ups, based on local rigidity arguments for some differential inclusions involving symmetrized gradients. This strategy allows us to establish the lower semicontinuity result even without a BD-analogue of Alberti's Rank-One Theorem in BV, which is not available at present. A similar strategy also makes it possible to give a proof of the classical lower semicontinuity theorem in BV without invoking Alberti's Theorem.