Higher integrability for constrained minimizers of integral functionals with ([formula omitted])-growth in low dimension

Abstract

We prove higher summability for the gradient of minimizers of strongly convex integral functionals of the Calculus of Variations ∫Ωf(x,Du(x))dx,u:Ω⊂Rn→SN−1,with growth conditions of (p,q)-type: |ξ|p≤f(x,ξ)≤C(|ξ|q+1),p<q,in low dimension. Our procedure is set in the framework of Fractional Sobolev Spaces and renders the desired regularity as the result of an approximation technique relying on estimates obtained through a careful use of difference quotients.