Schauder estimates for equations with cone metrics, I

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...

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 estimates for elliptic equations in Banach spaces associated with stochastic reaction-diffusion equations

We consider some reaction-diffusion equations perturbed by white noise and prove Schauder estimates for the elliptic problem associated with the generator of the corresponding transition semigroup, defined in the Banach space of continuous functions. This requires the proof of some new interpolation result. 2000 Mathematics Subject Classification AMS: 35R15, 60H15, 35B45

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 on Lie Groups