Regularity for Double Phase Problems at Nearly Linear Growth

Abstract

AbstractMinima of functionals of the type $$\begin{aligned} w\mapsto \int _{\varOmega }\left[ |Dw|\log (1+|Dw|)+a(x)|Dw|^{q}\right] \, \textrm{d}x, \quad 0\le a(\cdot ) \in C^{0, \alpha }, \end{aligned}$$ w ↦ ∫ Ω | D w | log ( 1 + | D w | ) + a ( x ) | D w | q d x , 0 ≤ a ( · ) ∈ C 0 , α , with $$\varOmega \subset {\mathbb {R}}^n$$ Ω ⊂ R n , have locally Hölder continuous gradient provided $$1< q < 1+\alpha /n$$ 1 < q < 1 + α / n .