Approximation of variational problems with a convexity constraint by PDEs of Abreu type

Guillaume Carlier (1, 2) , Teresa Radice (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales
3 Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”

Motivated by some variational problems subject to a convexity constraint, we consider an approximation using the logarithm of the Hessian determinant as a barrier for the constraint. We show that the minimizer of this penalization can be approached by solving a second boundary value problem for Abreu's equation which is a well-posed nonlinear fourth-order elliptic problem. More interestingly, a similar approximation result holds for the initial constrained variational problem.

Mots clés: Monge-Ampère operator; Abreu equation; Calculus of variations with a convexity constraint

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Guillaume Carlier, Teresa Radice. Approximation of variational problems with a convexity constraint by PDEs of Abreu type. Calculus of Variations and Partial Differential Equations, 2019. ⟨hal-01802925⟩