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<p>In this paper, we study interior estimates for solutions to linearized Monge–Ampère equations in divergence form with drift terms and the right-hand side containing the divergence of a bounded vector field. Equations of this type appear in the study of semigeostrophic equations in meteorology and the solvability of singular Abreu equations in the calculus of variations with a convexity constraint. We prove an interior Harnack inequality and Hölder estimates for solutions to equations of this type in two dimensions, and under an integrability assumption on the Hessian matrix of the Monge–Ampère potential in higher dimensions. Our results extend those of Le (Graduate studies in mathematics, vol 240, American Mathematical Society, 2024) to equations with drift terms.</p>

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Interior Harnack inequality and Hölder estimates for linearized Monge–Ampère equations in divergence form with Drift

  • 期刊论文

  • 发表日期：

    2025年3月24日

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  • Young Ho Kim

摘要

In this paper, we study interior estimates for solutions to linearized Monge–Ampère equations in divergence form with drift terms and the right-hand side containing the divergence of a bounded vector field. Equations of this type appear in the study of semigeostrophic equations in meteorology and the solvability of singular Abreu equations in the calculus of variations with a convexity constraint. We prove an interior Harnack inequality and Hölder estimates for solutions to equations of this type in two dimensions, and under an integrability assumption on the Hessian matrix of the Monge–Ampère potential in higher dimensions. Our results extend those of Le (Graduate studies in mathematics, vol 240, American Mathematical Society, 2024) to equations with drift terms.