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

TL;DR

This paper develops a generalized Schauder theory for degenerate and singular parabolic equations, enabling higher regularity results by approximating coefficients with s-polynomials, and connects to Monge--Ampère equations.

Contribution

It introduces a novel generalized Schauder theory using s-polynomials for degenerate/singular parabolic equations, extending classical methods to more complex operators.

Findings

Established $C_s^{2+eta}$ regularity for solutions.

Unified approach applicable to degenerate and singular equations.

Reproduces classical Schauder estimates and extends to new operators.

Abstract

In this paper, we study generalized Schauder theory for the degenerate/singular parabolic equations of the form $$u_t = a^{i'j'}u_{i'j'} + 2 x_n^{\gamma/2} a^{i'n} u_{i'n} + x_n^{\gamma} a^{nn} u_{nn} + b^{i'} u_{i'} + x_n^{\gamma/2} b^n u_{n} + c u + f \quad (\gamma \leq1).$$ When the equation above is singular, it can be derived from Monge--Amp\`ere 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 method to obtain higher regularity.