Schauder-type estimates for higher-order parabolic SPDEs

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 Sa...

Schauder and Lp Estimates for Parabolic Systems via Campanato Spaces

The following note deals with classical Schauder and L estimates in the setting of parabolic systems. For the heat equation these estimates are usually obtained via potential theoretic methods, i.e. by studying the fundamental solution (see e.g. [3], [8], and, for the elliptic case, [7]). For systems, however, it has become customary to base both Schauder and L theory on Campanato’s technique. ...

Blow-up Estimates for Higher-order Semilinear Parabolic Equations

Higher Order Derivative Estimates for Finite-difference Schemes for Linear Elliptic and Parabolic Equations

Notes on Schauder Estimates

Proof. Let g(x) = u(x) − sup∂Br(y) u − r2 − |x − y|2 2n supBr(y) f . We have ∆g = ∆u + supBr(y) f ≥ − f + supBr(y) f ≥ 0, that is, g is subharmonic in Br(y). Then supBr(y) g = sup∂Br(y) g = 0, so g ≤ 0 in Br(y) and the lemma follows. Lemma 2. If u is a solution to ∆u = f in Br(y) and v solves ∆v = 0 and v = u on ∂Br(y), then r2 − |x − y|2 2n inf Br(y) f ≤ v(x) − u(x) ≤ r 2 − |x − y|2 2n sup Br(...