Besov regularity for solutions of p-harmonic equations

Abstract

We establish the higher fractional differentiability of the solutions to nonlinear elliptic equations in divergence form, i.e., div⁡𝒜⁢(x,D⁢u)=div⁡F,{\operatorname{div}\mathcal{A}(x,Du)=\operatorname{div}F,} when 𝒜{\mathcal{A}} is a p-harmonic type operator, and under the assumption that x↦𝒜⁢(x,ξ){x\mapsto\mathcal{A}(x,\xi\/)} belongs to the critical Besov–Lipschitz space Bn/α,qα{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 div⁡F=0{\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↦𝒜⁢(x,ξ){x\mapsto\mathcal{A}(x,\xi\/)}.

Keywords

• nonlinear elliptic equations
• higher order fractional differentiability
• besov spaces
• triebel–lizorkin spaces
• 35b65
• 35j60
• 42b37
• 49n60