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Abstract We revisit the problem of approximating minimizers of certain convex functionals subject to a convexity constraint by solutions of fourth order equations of Abreu type. This approximation problem was studied in previous articles of Carlier–Radice (Approximation of variational problems with a convexity constraint by PDEs of Abreu type. Calc. Var. Partial Differential Equations 58 (2019), no. 5, Art. 170) and the author (Singular Abreu equations and minimizers of convex functionals with a convexity constraint, arXiv:1811.02355v3, Comm. Pure Appl. Math. , to appear), under the uniform convexity of both the Lagrangian and constraint barrier. By introducing a new approximating scheme, we completely remove the uniform convexity of both the Lagrangian and constraint barrier. Our analysis is applicable to variational problems motivated by the original 2D Rochet–Choné model in the monopolist's problem in Economics, and variational problems arising in the analysis of wrinkling patterns in floating elastic shells in Elasticity.
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This content will become publicly available on November 19, 2026
On approximation of convex functionals with a convexity constraint and general Lagrangians
In this note, we prove that minimizers of convex functionals with a convexity constraint and a general class of Lagrangians can be approximated by solutions to fourth-order Abreu-type equations. Our result generalizes that of Le (Twisted Harnack inequality and approximation of variational problems with a convexity constraint by singular Abreu equations.Adv. Math.434(2023)) where the case of quadratically growing Lagrangians was treated.
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- Award ID(s):
- 2054686
- PAR ID:
- 10649315
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Proceedings of the Royal Society of Edinburgh: Section A Mathematics
- ISSN:
- 0308-2105
- Page Range / eLocation ID:
- 1 to 17
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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