Global gradient estimates for spherical quasi-minimizers of integral functionals with p(x)-growth. (December 2018) - Byun, Sun-Sig; Ok, Jihoon; Youn, Yeonghun
Abstract: We investigate spherical quasi-minimizers (or so-called spherical Q -minimizers) u for integral functionals with p ( x ) -growth of F ( u, Ω ) ≔ ∫ Ω f ( x, D u ) − | G | p ( x ) − 2 G ⋅ D u d x . In particular, we find minimal regularity requirements on the integrand f ( x, ξ ), the variable exponent p ( x ), the boundary ∂ Ω of the bounded domain Ω as well as Q under which | G | p ( ⋅ ) ∈ L q ( Ω ) ⟹ | D u | p ( ⋅ ) ∈ L q ( Ω ) for every q ≥ 1 with the desired global Calderón–Zygmund estimate.