Regularity results for solutions to a class of obstacle problems. (December 2021) - Grimaldi, Antonio Giuseppe
Abstract: In this paper we prove some regularity properties of solutions to variational inequalities of the form ∫ Ω 〈 A ( x, u, D u ), D ( φ − u ) 〉 d x ≥ ∫ Ω B ( x, u, D u ) ( φ − u ) d x, ∀ φ ∈ K ψ ( Ω ) . Here Ω is a bounded open set of R n, n ≥ 2, the function ψ : Ω → [ − ∞, + ∞ ), called obstacle, belongs to the Sobolev class W 1, p ( Ω ) and K ψ ( Ω ) = w ∈ W 1, p ( Ω ) : w ≥ ψ q.o. in Ω is the class of the admissible functions. First we establish a local Calderòn–Zygmund type estimate proving that the gradient of the solutions is as integrable as the gradient of the obstacle in the scale of Lebesgue spaces L p q, for every q ∈ ( 1, ∞ ), provided the partial map ( x, u ) ↦ A ( x, u, ξ ) is Hölder continuous and B ( x, u, ξ ) satisfies a suitable growth condition. Next, this estimate allows us to prove that a higher differentiability in the scale of Besov spaces of the gradient of the obstacle transfers to the gradient of the solutions.