Partial regularity of minimizers of p (x )-growth functionals with 1 < p (x ) < 2
Kunihiro Usuba · Bulletin of the London Mathematical Society · 2015
We prove partial regularity of minimizers u for p ( x ) -energy functionals of the following type: E ( u ) = ∫ Ω ( A i j α β ( x , u ) D α u i D β u j ) p ( x ) / 2 d x , assuming that A i j α β ( x , u ) and p ( x ) are sufficiently smooth and that p ( x ) is subquadratic. We prove that u ∈ C 0 , α ( Ω 0 ) for some α ∈ ( 0 , 1 ) and an open set Ω 0 ⊂ Ω with H m - γ 1 ( Ω - Ω 0 ) = 0 , where H s denotes the s-dimensional Hausdorff measure and γ 1 = inf Ω p ( x ) .