Higher differentiability for minimizers of variational obstacle problems with Orlicz growth

In this paper, we study a local higher differentiability of minimizers for variational obstacle problems with Orlicz growth min {integral(Omega) f(x, Dv(x))dx: v is an element of kappa(psi)(Omega)}, where kappa(psi)(O) is an admissible set constrained by an obstacle function psi(x). Here, the integrand is supposed to satisfy the Orlicz growth phi(vertical bar Dv vertical bar|) for f(xi)(x, xi) similar to phi'(vertical bar xi vertical bar) with 2< b(1) <= t phi'(t)/phi(t) <= b(2) < +infinity for any t >= 0. Under the relaxed assumption for the partial map g(x) = D(xi x)f(x, xi)/phi' (vertical bar xi vertical bar) is an element of L-loc(b2+2)(Omega), we prove a higher differentiability of gradients of local minimizers by assuming an extra differentiability and the boundedness of psi, based on the De Giorgi iteration and the so-called difference quotient method. To achieve the higher differentiability of minimizers, our key motivation is to relax assumption on g(x) is an element of L-loc(b2+2)(Omega) instead of g(x) is an element of L-loc(b2+2)(Omega) as in [3]; moreover, we also extend the functional with p-growth shown as in [4] to the setting of Orlicz growth. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.