Regularity for minimizers of a class of non-autonomous functionals with sub-quadratic growth

Abstract

Abstract We consider functionals of the form ℱ ⁢ ( v , Ω ) = ∫ Ω f ⁢ ( x , D ⁢ v ⁢ ( x ) ) ⁢ 𝑑 x , \mathcal{F}(v,\Omega)=\int_{\Omega}f(x,Dv(x))\,dx, with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x variable belongs to a suitable Sobolev space W 1 , q {W^{1,q}} . We prove a higher differentiability result for the minimizers. We also infer a Lipschitz regularity result of minimizers if q > n {q>n} , and a result of higher integrability for the gradient if q = n {q=n} . The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.