Band 6, Heft 2

Abstract. In the present paper we consider the Dirichlet problem associated with a nonlinear singular elliptic equation whose differential operator arises in the level set formulation of the inverse mean curvature flow; namely, we study We introduce a suitable concept of weak solution, for which we prove existence and uniqueness of the homogeneous Dirichlet problem in a bounded open set of ℝ N for data f belonging to suitable Lebesgue spaces. Moreover, examples of explicit solutions are shown.

Abstract. We consider a variational model for damaged elastic materials. This model depends on three small parameters, which are related to the cost of the damage, to the width of the damaged regions, and to the minimum elasticity constant attained in the damaged regions. As these parameters tend to zero, our models Γ-converge to a model for brittle fracture, for fracture with a cohesive zone, or for perfect plasticity, depending on the asymptotic ratios of the three parameters.

Abstract. In the present work we establish an energy quantization (or energy identity) result for solutions to scaling invariant variational problems in dimension 4 which includes biharmonic maps (extrinsic and intrinsic). To that end we first establish an angular energy quantization for solutions to critical linear 4th order elliptic systems with antisymmetric potentials. The method is inspired by the one introduced by the authors previously in “Angular energy quantization for linear elliptic systems with antisymmetric potentials and applications” (2011) for 2nd order problems.

Abstract. Relaxation theorems for multiple integrals on , where , are proved under general conditions on the integrand which is Borel measurable and not necessarily finite. We involve a localization principle that we previously used to prove a general lower semicontinuity result. We apply these general results to the relaxation of nonconvex integrals with exponential-growth.