Partial regularity of minimizers of p(x)-growth functionals with 1 < p(x) < 2

usuba, kunihiro;

We prove partial regularity of minimizers u for p(x)-energy functionals of the following type:
E(u) =Ω(Aαβij (x, u)DαuiDβuj)p(x)/2 dx, assuming that Aαβ
ij (x, u) and p(x) are sufficiently smooth and that p(x) is subquadratic.
We prove that u ∈ C0,α(Ω0) for some α ∈ (0,1) and an open set Ω0
⊂Ω with Hm−γ1(Ω − Ω0) = 0, where Hs denotes the s-dimensional Hausdorff measure and γ1 = infΩ p(x).