Abstract

This talk considers minimization problems for integral functionals on the linear-growth space BV of functions of bounded variation and on the space BD of functions of bounded deformation. The space BD 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 integral functionals with linear growth on the space BD under the (natural) assumption of symmetric-quasiconvexity. This establishes the existence of solutions to a class of minimization problems in which fractal phenomena may occur. The proof proceeds via generalized Young measures and a construction of good blow-ups, based on local rigidity/ellipticity arguments for some differential inclusions. A similar strategy also allows to give a proof of the classical lower semicontinuity theorem in BV for quasiconvex integral functionals without invoking Alberti’s Rank-One Theorem.