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Generalized Schauder theory and its application to degenerate/singular parabolic equations

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In this paper, we study generalized Schauder theory for the degenerate/singular parabolic equations of the form

$$\begin{aligned} u_t = a^{i'j'}u_{i'j'} + 2 x_n^{\gamma /2} a^{i'n} u_{i'n} + x_n^{\gamma } a^{nn} u_{nn} + b^{i'} u_{i'} + x_n^{\gamma /2} b^n u_{n} + c u + f \quad (\gamma \le 1). \end{aligned}$$

When the equation above is singular, it can be derived from Monge–Ampère equations by using the partial Legendre transform. Also, we study the fractional version of Taylor expansion for the solution u, which is called s-polynomial. To prove \(C_s^{2+\alpha }\)-regularity and higher regularity of the solution u, we establish generalized Schauder theory which approximates coefficients of the operator with s-polynomials rather than constants. The generalized Schauder theory not only recovers the proof for uniformly parabolic equations but is also applicable to other operators that are difficult to apply the bootstrap argument to obtain higher regularity.

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Correspondence to Hyungsung Yun.

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Ki-Ahm Lee was supported by the National Research Foundation of Korea (NRF), Ministry of Science and ICT, South Korea, Grant: NRF-2020R1A2C1A01006256. Hyungsung Yun was supported by a KIAS Individual Grant (no. MG097801) at Korea Institute for Advanced Study.

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Kim, T., Lee, KA. & Yun, H. Generalized Schauder theory and its application to degenerate/singular parabolic equations. Math. Ann. (2024).

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