Generalized Schauder Theory and its Application to Degenerate/Singular Parabolic Equations

2023/04/18 by Takwon Kim, Kim, Takwon, Ki-Ahm Lee +3 · 1 citation
Computer Science · Mathematics · #35B65 #35K65 #35K67 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2304.08734

openalex publication_date 2023/04/18 · openalex created_date 2023/04/22 · openalex updated_date 2026/08/01

Abstract

In this paper, we study generalized Schauder theory for the degenerate/singular parabolic equations of the form u_t = a^i'j'u_i'j' + 2 x_n^γ/2 a^i'n u_i'n + x_n^γ aⁿⁿ u_nn + b^i' u_i' + x_n^γ/2 bⁿ u_n + c u + f (γ≤1). 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+α-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 method to obtain higher regularity.

Cited by

• Boundary Hölder gradient estimates for fully nonlinear degenerate or singular parabolic equations

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