Regularity results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions. (December 2022) - Gentile, Andrea; Giova, Raffaella
Abstract: We establish some higher differentiability results for solution to non-autonomous obstacle problems of the form min ∫ Ω f x, D v ( x ) d x : v ∈ K ψ ( Ω ), where the function f satisfies p − growth conditions with respect to the gradient variable, for 1 < p < 2, and K ψ ( Ω ) is the class of admissible functions. Here we show that, if the obstacle ψ is bounded, then a Sobolev regularity assumption on the gradient of the obstacle ψ transfers to the gradient of the solution, provided the partial map x ↦ D ξ f ( x, ξ ) belongs to a Sobolev space, W 1, p + 2 . The novelty here is that we deal with subquadratic growth conditions with respect to the gradient variable, i.e. f ( x, ξ ) ≈ a ( x ) | ξ | p with 1 < p < 2, and where the map a belongs to a Sobolev space.