Higher differentiability of solutions to a class of obstacle problems under non-standard growth conditions

Abstract

Abstract We establish the higher differentiability of integer order of solutions to a class of obstacle problems assuming that the gradient of the obstacle possesses an extra integer differentiability property. We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality of the form ∫ Ω 〈 𝒜 ⁢ ( x , D ⁢ u ) , D ⁢ ( φ - u ) 〉 ⁢ 𝑑 x ≥ 0 for all ⁢ φ ∈ 𝒦 ψ ⁢ ( Ω ) . \int_{\Omega}\langle\mathcal{A}(x,Du),D(\varphi-u)\rangle\,dx\geq 0\quad\text{% for all }\varphi\in\mathcal{K}_{\psi}(\Omega). The main novelty is that the operator 𝒜 {\mathcal{A}} satisfies the so-called p , q {p,q} -growth conditions with p and q linked by the relation q p < 1 + 1 n - 1 r , \frac{q}{p}<1+\frac{1}{n}-\frac{1}{r}, for r > n {r>n} . Here ψ ∈ W 1 , p ⁢ ( Ω ) {\psi\in W^{1,p}(\Omega)} is a fixed function, called obstacle, for which we assume D ⁢ ψ ∈ W loc 1 , 2 ⁢ q - p ⁢ ( Ω ) {D\psi\in W^{1,2q-p}_{\mathrm{loc}}(\Omega)} , and 𝒦 ψ = { w ∈ W 1 , p ⁢ ( Ω ) : w ≥ ψ ⁢ a.e. in ⁢ Ω } {\mathcal{K}_{\psi}=\{w\in W^{1,p}(\Omega):w\geq\psi\text{ a.e. in }\Omega\}} is the class of admissible functions. We require for the partial map x ↦ 𝒜 ⁢ ( x , ξ ) {x\mapsto\mathcal{A}(x,\xi\/)} a higher differentiability of Sobolev order in the space W 1 , r {W^{1,r}} , with r > n {r>n} satisfying the condition above.