On W1,(⋅)-estimates for elliptic equations of p(x)-Laplacian type

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...

The Laplacian, Second Order Elliptic Equations

The Nehari Manifold for a Class of Elliptic Equations of P-laplacian Type

1 1 , , 0, , r s p u u h x u dx g x u dx x u x + + −∆ = + ∈ Ω = ∈ ∂Ω () 1, 0 p W Ω () () () 1 1 , 0. r s p u x h x u dx g x u dx in u on + +  −∆ = + Ω   = ∂Ω   () E () () 1 1 / r p s Np N p N p < < − < < − + − () () () 0 0 r h L L C ∞ ∈ Ω Ω Ω   0 1 1 1, r r p * + + = ()() 0. 1 Np r Np r N p = − + − () () 0 s g L L ∞ ∈ Ω Ω  0 1 1 1, s s p * + + = ()() 0 ,. 1 Np Np s p Np s N p N p *   ...

Quasilinearization Technique for Φ-Laplacian Type Equations

Copyright q 2012 I. Yermachenko and F. Sadyrbaev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. An equation d/dt Φ t, x′ f t, x 0 is considered together with the boundary conditions Φ a, x′ a 0, x b 0. This problem under appropr...

Discrete/continuous elliptic Harnack inequality and kernel estimates for functions of the Laplacian on a graph

This paper introduces certain elliptic Harnack inequalities for harmonic functions in the setting of the product space M×X, where M is a (weighted) Riemannian manifold and X is a countable (symmetrically weighted) graph. Since some standard arguments for the elliptic case fail in this “mixed” setting, we adapt ideas from the discrete parabolic case found in Delmotte, 1999. We then present some ...