2024/05/22 by Wang, Ling
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2405.13297
In this paper, we investigate the interior Hölder regularity of solutions to the linearized Monge-Ampère 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-Gutiérrez Hölder estimate (Amer. J. Math. 119 (1997), no. 2, 423-465) and the result of Le (Comm. Math. Phys. 360 (2018), no. 1, 271-305) for the linearized Monge-Ampère 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-Ampère equation. Building on this idea, we also establish a new Moser-Trudinger type inequality in dimension two. In higher dimensions, we derive the interior Hölder estimate under certain integrability assumptions on the coefficients using De Giorgi's iteration.