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Global \(C^{1+\alpha ,\frac{1+\alpha }{2}}\) Regularity on the Linearized Parabolic Monge–Ampère Equation

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Abstract

In this paper, we establish global \(C^{1+\alpha ,\frac{1+\alpha }{2}}\) estimates for solutions of the linearized parabolic Monge–Ampère equation

$$\begin{aligned} {\mathcal {L}}_\phi u(x,t):=-u_t\,\mathrm {det}D^2\phi (x)+\mathrm {tr}[\Phi (x) D^2 u]=f(x,t) \end{aligned}$$

under appropriate conditions on the domain, Monge–Ampère measures, boundary data and f, where \(\Phi :=\mathrm {det}(D^2\phi )(D^2\phi )^{-1}\) is the cofactor of the Hessian of \(D^2\phi \).

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Funding

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

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Authors and Affiliations

  1. LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China

    Lin Tang

  2. School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, MOE, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Qian Zhang

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  1. Lin Tang
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  2. Qian Zhang
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Correspondence to Lin Tang.

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Tang, L., Zhang, Q. Global \(C^{1+\alpha ,\frac{1+\alpha }{2}}\) Regularity on the Linearized Parabolic Monge–Ampère Equation. Results Math 77, 224 (2022). https://doi.org/10.1007/s00025-022-01751-z

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