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An Campanato type regularity condition is established for a class of WX local minimisers $\overline{u}$ of the general variational integral ${\int }_{\Omega }F\left(\nabla u\left(x\right)\right)\mathrm{d}x$ where $\Omega \subset {ℝ}^n$ is an open bounded domain, is of class C, is strongly quasi-convex and satisfies the growth condition$F\left(\xi \right)\le c(1+|\xi {|}^p)$ for a and where the corresponding Banach spaces X are the Morrey-Campanato space ${ℒ}^{p,\mu }(\Omega ,{ℝ}^{N\times n})$, < , Campanato space ${ℒ}^{p,n}(\Omega ,{ℝ}^{N\times n})$ and the space of bounded mean oscillation $\mathrm{BMO}\Omega ,{ℝ}^{N\times n})$. The admissible maps $u:\Omega \to {ℝ}^N$ are of Sobolev class W, satisfying a Dirichlet boundary condition, and to help...