Besov regularity for solutions of p-harmonic equations

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