Higher differentiability of solutions for a class of obstacle problems with non standard growth conditions

We are going to prove a higher differentiability result for the solutions to a class of variational obstacle problems of the formmin{integral F-omega(x, D omega)dx : omega is an element of kappa(psi)(omega)}where the function F satisfies non standard growth conditions of the type nu|z|p <= F(x, z) <= L(1 + |z|(q)) with 1 < p < q and 0 < nu < L; in particular we will consider the subquadratic case: 1 < p < q < 2.Moreover omega subset of R-n, n > 2 is a bounded open domain, the function psi : omega -> [-infinity, +infinity) is called obstacle and it belongs to the Sobolev class W-1,W-p(omega), p > 1 and kappa(psi)(12) is the set of admissible functions, i.e.kappa(psi)(omega) = {omega is an element of u(0) +W-0( )1,p(omega) :omega >psi a.e. in omega}where u0 is a fixed boundary value. The main result consists in showing thatD psi is an element of W-loc(1,r)(omega) -> (1 + |Du|(2)) p-2/4 Du is an element of W-loc(1,2 )(omega)under the conditions:q/p < 1 < 1/n - 1/r, 1 < p < q < 2 < n < r. (c) 2022 Elsevier Inc. All rights reserved.