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Interior \(C^{1,\alpha }\) regularity for the linearized Monge–Ampère equation with VMO type coefficients

  • Lin TangEmail author
  • Qian Zhang
Original Paper
First Online:

Abstract

In this paper, we establish interior \(C^{1,\alpha }\) estimates for solutions of the linearized Monge–Ampère equation
$$\begin{aligned} {\mathcal {L}}_{\phi }u:=\text {tr}[\varPhi D^2 u]=f, \end{aligned}$$
where the density of the Monge–Ampère measure \(g:=\text {det}D^2\phi \) satisfies a VMO-type condition and \(\varPhi :=(\text {det}D^2\phi )(D^2\phi )^{-1}\) is the cofactor matrix of \(D^2\phi \).

Keywords

Linearized Monge–Ampère equation \(C^{1, \alpha }\) estimates VMO-type condition 

Mathematics Subject Classification

35J96 35J60 35B45 
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Notes

Acknowledgements

The research was supported by the NNSF (11771023) of China.

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Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesPeking UniversityBeijingPeople’s Republic of China

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