Schauder Estimates for Elliptic and Parabolic Equations

The Schauder estimate for the Laplace equation was traditionally built upon the Newton potential theory. Different proofs were found later by Campanato [Ca], in which he introduced the Campanato space; Peetre [P], who used the convolution of functions; Trudinger [T], who used the mollification of functions; and Simon [Si], who used a blowup argument. Also a perturbation argument was found by Safonov [S1,S2] and Caffarelli [C1, CC] for fully nonlinear uniformly elliptic equations, which also applies to the Laplace equation. In this note we give an elementary and simple proof for the Schauder estimates for elliptic and parabolic equations. Our proof allows the right hand side to be Dini continuous and also give a sharp estimate for the modulus of continuity of the second derivatives. It also yields the log-Lipschitz continuity of the gradient for equations with bounded right hand side. Moreover, it also applies to nonlinear equations.