Homogenization of Periodic Finsler Metrics

We prove an homogenization result in W¹,¹ and in BV for a sequence $(F_\varepsilon)$ of functionals of the form $F_\varepsilon(u) = \int_0^1 f(u/\varepsilon,u')\,dt$ where ε is a positive parameter which tends to zero, f : ℝⁿ × ℝⁿ → [0, +∞) is [0, 1)ⁿ-periodic in the first variable, convex in the second variable and satisfies a suitable growth condition of order one. Under the additional assumption that f(x, ·) is positively 1-homogeneous, we show how our result is equivalent to the analogous homogenization result (dealt with by Acerbi and Buttazzo) in which growth conditions of order p > 1 are considered.