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

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  19 November 2025

Young Ho Kim Open the ORCID record for Young Ho Kim [Opens in a new window]
Young Ho Kim*
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas, United States Department of Mathematics, Indiana University, Bloomington, Indiana, United States (yhkim@tamu.edu)
*
*Corresponding author.
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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 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.

Keywords

Convex functionalconvexity constraintlinearized Monge–Ampère equationMonge–Ampère equationsingular Abreu equation

MSC classification

Primary: 35J35: Variational methods for higher-order elliptic equations
Secondary: 35J40: Boundary value problems for higher-order elliptic equations 35J96: Elliptic Monge-Ampère equations

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 17
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.

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