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.

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关于用不具有一致凸性的奇异 Abreu 方程逼近具有凸性约束的凸泛函的极小值

我们通过 Abreu 类型的四阶方程的解来重新审视受凸性约束的某些凸泛函的逼近极小化问题。在 Carlier-Radice 之前的文章中研究了这个近似问题（Abreu 类型 PDE 对具有凸性约束的变分问题的近似。计算。变量。偏微分方程58（2019 年），没有。5，艺术。170）和作者（奇异 Abreu 方程和具有凸性约束的凸泛函的最小化器，arXiv：1811.02355v3，通讯。纯应用。数学。, 出现），在拉格朗日和约束障碍的一致凸性下。通过引入一种新的近似方案，我们完全消除了拉格朗日和约束障碍的一致凸性。我们的分析适用于经济学中垄断者问题中由原始 2D Rochet-Choné 模型激发的变分问题，以及弹性中浮动弹性壳的起皱模式分析中出现的变分问题。