Global integrability for minimizers of anisotropic functionals

We consider integral functionals in which the density has growth p _i with respect to ${frac{partial u}{partial x_i}}$ , like in $$intlimits_{Omega}left( left| frac{partial u}{partial x_1}(x) right|^{p_1} + left|frac{partial u}{partial x_2}(x)right|^{p_2} + cdots + left|frac{partial u}{partial x_n}(x) right|^{p_n} right) dx.$$ We show that higher integrability of the boundary datum forces minimizer to be more integrable.