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

Carlier, Guillaume; Radice, Teresa (2019), Approximation of variational problems with a convexity constraint by PDEs of Abreu type, Calculus of Variations and Partial Differential Equations, 58, p. 16. 10.1007/s00526-019-1613-1

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Type
Article accepté pour publication ou publié
Date
2019
Journal name
Calculus of Variations and Partial Differential Equations
Volume
58
Pages
16
Publication identifier
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Author(s)
Carlier, Guillaume
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Radice, Teresa
Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”
Abstract (EN)
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.
Subjects / Keywords
Abreu equation; Monge-Ampère operator; calculus of varia-; tions with a convexity constraint