Periodic solutions and homogenization of non linear variational problems

Abstract

We prove an homogenization formula for some non linear variational problems that extends the analogous one known in the linear case. Namely the solution ue of the problem $$\int\limits_\Omega {\left\{ {f\left( {\frac{x}{\varepsilon },Du_\varepsilon } \right) - \varphi \left( x \right)u_\varepsilon } \right\}dx} = minimum,$$ where the boundary data are independent of ɛ and f=f(x, ξ) is periodic in x, converges in some Lp space as ɛ goes to zero to the solution of an analogous variational problem whose integrand can be evaluated from f.