Global Integrability for Solutions to Obstacle Problems (Journal of Partial Differential Equations, 2022)
Denote $$ {\cal K} _{ψ, θ} (Ω) =\left\{ v\in W^{1,p} (Ω) : v\ge ψ, a.e. and v-θ \in W_0^{1,p} (Ω) \right\}, $$ where $ψ$ is any function in $Ω \subset \mathbb R^N$, $N\ge 2$, with values in $\mathbb R \cup \{± ∞\}$ and $θ $ is a measurable function. This paper deals with global integrability for $u \in {\cal K}_{ψ, θ}$ such that$$ \int_Ω \langle {\cal A} (x, abla u), abla (w-u) \rangle {\rm d}x \ge \int_Ω \langle F, abla (w-u) \rangle {\rm d}x, \ \ \forall\ w \in {\cal K}_{ψ,θ} (Ω), $$ with $|{\cal A} (x,ξ)| ≈ |ξ| ^{p-1}$, $1