Schauder estimates for equationswith fractional derivatives

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(...

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 Estimates on Lie Groups

Uniform Schauder Estimates for Regularized Hypoelliptic Equations

In this paper we are concerned with a family of elliptic operators represented as sum of square vector fields: L = ∑m i=1X 2 i + ∆ in Rn, where ∆ is the Laplace operator, m < n, and the limit operator L = ∑m i=1X 2 i is hypoelliptic. Here we establish Schauder’s estimates, uniform with respect to the parameter , of solution of the approximated equation L u = f , using a modification of the lift...

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. ...