Partial regularity of solutions to p(x)-Laplacian PDEs with discontinuous coefficients

For Ω⊆Rn an open and bounded region we consider solutions u∈Wloc 1,p(x)(Ω;RN), with N>1, of the p(x)-Laplacian system ∇⋅(a(x)|Du|p(x)−2Du)=0, a.e. x∈Ω, where concerning the coefficient function x↦a(x) we assume only that a∈W1,q(Ω)∩L∞(Ω), where q>1 is essentially arbitrary. This implies that the coefficient in the PDE can be highly irregular, and yet in spite of this we still recover that u∈Cloc 0,α(Ω0), for each 0<α<1, where Ω0⊆Ω is a set of full measure. Due to the variational methodology that we employ, our results apply to the more general question of the regularity of the integral functional ∫Ωa(x)|Du|p(x)dx. © 2019 Elsevier Inc.