Gradient estimates for multi-phase problems in Campanato spaces

We establish a new Campanato type estimate for the weak solutions of a class of multi-phase problems. The problem under consideration is characterized by the fact that both ellipticity and growth switch between three different types of polynomial according to the position, which describes a feature of strongly anisotropic materials. The results obtained in this paper are different from the BMO type estimates for the usual p-Laplacian equation due to DiBenedetto and Manfredi. The content of this paper is in close relationship with the recent pioneering contributions of Marcellini and Mingione in the qualitative analysis of multi-phase problems.