Interior H\"older regularity of the linearized Monge-Amp\`ere equation

Ling Wang

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arXiv · 2405.13297

Interior H\"older regularity of the linearized Monge-Amp\`ere equation

Ling Wang

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Abstract

In this paper, we investigate the interior H\"older regularity of solutions to the linearized Monge-Amp\`ere equation. In particular, we focus on the cases with singular right-hand side, which arise from the study of the semigeostrophic equation and singular Abreu equations. In the two-dimensional case, we give a new proof of the Caffarelli-Guti\'errez H\"older estimate (\textit{Amer. J. Math.} \textbf{119} (1997), no.\,2, 423-465) and the result of Le (\textit{Comm. Math. Phys.} \textbf{360} (2018), no.\,1, 271-305) for the linearized Monge-Amp\`ere equation with singular right-hand side term in divergence form. The main new ingredient in the proof contains the application of the partial Legendre transform to the linearized Monge-Amp\`ere equation. Building on this idea, we also establish a new Moser-Trudinger type inequality in dimension two. In higher dimensions, we derive the interior H\"older estimate under certain integrability assumptions on the coefficients using De Giorgi's iteration.

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Ling Wang. 2024-05-22. Interior H\"older regularity of the linearized Monge-Amp\`ere equation. https://arxiv.org/abs/2405.13297

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