Pure mathematics

Laplace's equation

Materials science

Planar

Computer graphics (images)

Plane (geometry)

Range (aeronautics)

Economics

Computer science

Harmonic function

Mathematics

Laplace transform

Mathematical analysis

Geometry

Composite material

Partial differential equation

Finance

Complex plane

Order (exchange)

Journal of Differential Equations

The paper is concerned with higher order Calderón-Zygmund estimates for the p-Laplace equation−div(A(∇u)):=−div(|∇u|p−2∇u)=−divF,1<p<∞. We are able to transfer local interior Besov and Triebel-Lizorkin regularity up to first order derivatives from the force term F to the flux A(∇u). For p≥2 we show that F∈Bϱ,qs implies A(∇u)∈Bϱ,qs for any s∈(0,1) and all reasonable ϱ,q∈(0,∞] in the planar case. The result fails for p<2. In case of higher dimensions and systems we have a smallness restriction on s. The quasi-Banach case 0<min⁡{ϱ,q}<1 is included, since it has important applications in the adaptive finite element analysis. As an intermediate step we prove new linear decay estimates for p-harmonic functions in the plane for the full range 1<p<∞.

Higher order Calderón-Zygmund estimates for the p-Laplace equation