Besov regularity for the elliptic p-harmonic equations in the non-quadratic case

Abstract

In this article, we mainly establish the local extra fractional differentiability (Besov regularity) of weak solutions for the following divergence nonlinear elliptic equations of pp-Laplacian type: divA(Du,x)=divF,\hspace{0.1em}\text{div}\hspace{0.1em}A(Du,x)=\hspace{0.1em}\text{div}\hspace{0.1em}{\bf{F}}, where AA is a Carathéodory function with p−1p-1 growth for 1<p<21\lt p\lt 2. The standard example for the aforementioned equations is the classical elliptic pp-Laplacian equation div(∣Du∣p−2Du)=divF,for1<p<2.\hspace{0.1em}\text{div}\hspace{0.1em}({| Du| }^{p-2}Du)=\hspace{0.1em}\text{div}\hspace{0.1em}{\bf{F}},\hspace{1.0em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}1\lt p\lt 2. Remarkably, the case 1<p<21\lt p\lt 2 is very different from the case p≥2p\ge 2 since the modulus of ellipticity in the elliptic pp-Laplacian equation tends to infinity when ∣Du∣→0| Du| \to 0.

Keywords

• besov
• fractional
• regularity
• p-laplacian
• elliptic
• weak solutions
• 35j60
• 35j70