Besov regularity for solutions of p-harmonic equations

Albert Clop,Antonia Passarelli Di Napoli,Raffaella Giova

doi:10.1515/anona-2017-0030

Abstract

Abstract We establish the higher fractional differentiability of the solutions to nonlinear elliptic equations in divergence form, i.e., {\operatorname{div}\mathcal{A}(x,Du)=\operatorname{div}F,} when {\mathcal{A}} is a p-harmonic type operator, and under the assumption that {x\mapsto\mathcal{A}(x,\xi\/)} belongs to the critical Besov–Lipschitz space {B^{\alpha}_{{n/\alpha},q}} . We prove that some fractional differentiability assumptions on F transfer to Du with no losses in the natural exponent of integrability. When {\operatorname{div}F=0} , we show that an analogous extra differentiability property for Du holds true under a Triebel–Lizorkin assumption on the partial map {x\mapsto\mathcal{A}(x,\xi\/)} .

Highlights

In this paper we study the extra fractional differentiability of weak solutions of the following nonlinear elliptic equations in divergence form: div A(x, Du) = div F in Ω, (1.1)where Ω ⊂ Rn, n ≥ 2, is a domain, u : Ω → R, F : Ω → Rn, and A : Ω × Rn → Rn is a Carathéodory function with p − 1 growth
We prove that some fractional differentiability assumptions on F transfer to Du with no losses in the natural exponent of integrability
When div F = 0, we show that an analogous extra differentiability property for Du holds true under a Triebel–Lizorkin assumption on the partial map x 󳨃→ A(x, ξ )

Summary

Introduction

Recent developments show that the Lipschitz regularity of the partial map x → A(x, ⋅ ) can be weakened in a W1,n assumption on the coefficients, both in the linear and in the nonlinear setting, in order to get higher differentiability of the solution of integer order. The ellipticity assumption (A1) is lost when |ξ | approaches zero, and the estimates worsen even in the classical theory (see [22]) In this case we are not able to prove an extra fractional differentiability of the function Vp(Du) completely analogous to our previous theorem. For proving Theorems 1.1, 1.2 and 1.3, we shall use a result proved in [1] (see Theorem 2.5 in Section 2.2 below), and combine It with the Sobolev type embedding for Besov Lipschitz spaces to obtain the higher integrability of the gradient of the solutions of equation (1.1).

Notations and preliminary results

Besov–Lipschitz spaces

VMO coefficients in Rn

Difference quotient