Global Schauder Estimates for the $$\mathbf {p}$$-Laplace System

Eigenvalues Estimates for the p-Laplace Operator on Manifolds

The Laplace-Beltrami operator on a Riemannian manifold, its spectral theory and the relations between its first eigenvalue and the geometrical data of the manifold, such as curvatures, diameter, injectivity radius, volume, has been extensively studied in the recent mathematical literature. In the last few years, another operator, called p-Laplacian, arising from problems on Non-Newtonian Fluids...

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...