Regularity results for solutions to obstacle problems with Sobolev coefficients

[摘要] We establish the higher differentiability of solutions to a class of obstacle problems of the type min {(Omega)integral f (x, Dv(x))dx : v epsilon K (psi)(Omega)}, where psi is a fixed function called obstacle, K-psi (Omega) = {v epsilon W-loc(1,p) (Omega, R ) : v >= psi a.e. in Omega} and the convex integrand f satisfies p-growth conditions with respect to the gradient variable. We derive that the higher differentiability property of the weak solution v is related to the regularity of the assigned psi , under a suitable Sobolev assumption on the partial map x bar right arrow D-xi f (x, xi) . The main novelty is that such assumption is independent of the dimension n and this, in the case p <= n- 2, allows us to manage coefficients in a Sobolev class below the critical one W 1 ,n . (c) 2020 Elsevier Inc. All rights reserved.

[发布日期] 2020-11-05

[关键词] Local minimizers;Obstacle problems;Higher differentiability;Sobolev coefficients