Duality Theory and Applications to Unilateral Problems

New results on infinite dimensional duality in elastic-plastic torsion

“…The theorem means that, if we consider the solution u to variational inequality (9), then conditions (10) are satisfied. From conditions (10) it follows that, if u belongs to the elastic region E, µ ≡ 0 and then u is a solution of the elliptic equation Lu = f , and, in particular, the solution of (9) solves the elastic-plastic torsion problem. Conversely it is easily proved that, if u ∈ K satisfies conditions (10), then u solves variational inequality (9).…”

“…Recently in [10], among other results, the authors improve the result by Brezis, proving the existence of Lagrange multipliers for the following variational inequality, more general than (1): Find u ∈ K ∇ such that:…”

“…It is necessary to apply a new strong duality principle introduced in [8], that can be applied in many infinite dimensional settings. By means of this new theory in [10] the authors prove the existence of a L ∞ Lagrange multiplier for (5) assuming a constraint qualification condition (Assumption S), that is a necessary and sufficient condition in order that strong duality holds. The paper is organized in the following way: in Section 2 we present, with a short proof, the result of existence of a Radon measure, which plays the role of a Lagrange multiplier for the elastic-plastic torsion problem.…”

“…The paper is organized in the following way: in Section 2 we present, with a short proof, the result of existence of a Radon measure, which plays the role of a Lagrange multiplier for the elastic-plastic torsion problem. In Section 3 we resume the new strong duality theory introduced in [8] and the applications, contained in [10], of this theory to the Lagrange multipliers associated to the elastic-plastic torsion problem and finally in Section 4 we provide the conclusions.…”

Cybersecurity investments with nonlinear budget constraints and conservation laws: variational equilibrium, marginal expected utilities, and Lagrange multipliers

“…(), whereas for studies on the Lagrange theory and its application to variational models we refer to Daniele et al. (, , ), Giuffrè et al. (, ), Giuffrè and Maugeri (), and Toyasaki et al.…”