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arXiv · 2511.15621

A Green's function approach to linearized Monge-Amp\`ere equations in divergence form and application to singular Abreu type equations

Chong Gu·Nam Q. Le
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Abstract

In this paper, we establish local and global regularity estimates for linearized Monge-Amp\`ere equations in divergence form via critical Lorentz space estimates for the Green's function of the linearized Monge-Amp\`ere operator and its gradient. These estimates hold under suitable conditions on the data and the convex Monge-Amp\`ere potential is assumed to have Hessian determinant bounded between two positive constants. As an application, we obtain the solvability in all dimensions of the second boundary value problem for a class of singular fourth-order Abreu type equations that arise from the approximation analysis of variational problems subject to convexity constraints.

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Chong Gu, Nam Q. Le. 2025-11-19. A Green's function approach to linearized Monge-Amp\`ere equations in divergence form and application to singular Abreu type equations. https://arxiv.org/abs/2511.15621

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