On approximating minimizers of convex functionals with a convexity constraint by singular Abreu equations without uniform convexity

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 convexi...

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Bibliographic Details
Published in:Proceedings of the Royal Society of Edinburgh. Section A. Mathematics Vol. 151; no. 1; pp. 356 - 376
Main Author: Le, Nam Q.
Format: Journal Article
Language:English
Published: Edinburgh, UK Royal Society of Edinburgh Scotland Foundation 01.02.2021
Cambridge University Press
Subjects:
Approximation
Convexity
Economic analysis
Economic models
Elastic shells
Partial differential equations
Two dimensional models
Singular Abreu equation
Second boundary value problem
35J96
35J40
49K30
Convex functional
35B40
Wrinkling patterns
Rochet–Choné model
Convexity constraint
ISSN:0308-2105, 1473-7124
Online Access:Get full text
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Summary: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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ISSN:0308-2105
1473-7124
DOI:10.1017/prm.2020.18