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

Part of: Partial differential equations Optimality conditions Elliptic equations and systems

Published online by Cambridge University Press:  11 March 2020

Nam Q. Le
Nam Q. Le*
Affiliation:
Department of Mathematics, Indiana University, 831 E 3rd St, Bloomington, IN47405, USA (nqle@indiana.edu)
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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 Equations58 (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.

Keywords

Singular Abreu equationConvex functionalConvexity constraintSecond boundary value problemRochet–Choné modelWrinkling patterns

MSC classification

Primary: 49K30: Optimal solutions belonging to restricted classes
Secondary: 35B40: Asymptotic behavior of solutions 35J40: Boundary value problems for higher-order elliptic equations 35J96: Elliptic Monge-Ampère equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 1 , February 2021 , pp. 356 - 376
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Copyright
Copyright © The Author(s), 2020. Published by The Royal Society of Edinburgh

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