On approximation of convex functionals with a convexity constraint and general Lagrangians

TL;DR

This work addresses approximating minimizers of convex functionals with a convexity constraint by solutions of Abreu-type fourth-order equations in dimensions $n\ge 2$, for a broad class of Lagrangians $F(x,u,Du)$. The authors modify the approximation scheme with a convex penalty $G$ and a perturbed boundary data $\widetilde{\varphi}_\varepsilon$, replacing the prior quadratic growth requirement on $F$ and establishing solvability of the Abreu-type second boundary value problem in $W^{4,s}(\Omega)$. They prove uniform a priori estimates and use Leray–Schauder degree theory to obtain existence of solutions $u_\varepsilon$, then show that a subsequence converges uniformly on compact subsets to a minimizer of $J(u)=\int_{\Omega_0} F(x,u,Du)\,dx$ over $\overline{S}[\varphi,\Omega_0]$. The convergence argument ensures that the limit minimizes the original convex constrained variational problem, markedly extending prior results to general (non-quadratic) Lagrangians and higher dimensions. This provides a robust analytical framework for convexity-constrained variational problems, including economically motivated models such as Rochet-Choné.

Abstract

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 equations of Abreu type. 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.

Theorems & Definitions (19)

• Theorem 1.1
• Remark 1.2
• Proposition 2.1
• Lemma 2.2: Uniform $L^\infty$ bound on $u_\varepsilon$
• proof
• Remark 2.3
• Corollary 2.4
• Remark 2.5
• Lemma 2.6: Estimates for $f_\varepsilon$ in $\Omega_0$
• proof